{"id":35287,"date":"2024-12-20T13:00:16","date_gmt":"2024-12-20T13:00:16","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35287"},"modified":"2025-12-11T17:08:40","modified_gmt":"2025-12-11T17:08:40","slug":"theorema-weierstrassii-de-valoribus-extremis","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/","title":{"rendered":"THEOREMA WEIERSTRASSII DE VALORIBUS EXTREMIS"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>Theorema Weierstrassii de Valoribus Extremis<\/h1>\n<p style=\"text-align:center;\"><em>Cur in tot problematis optimizationis fere pro certo habetur \u201cmaximum exsistere\u201d aut \u201csemper minimum dari\u201d in quodam intervallo, cum re vera nihil cogat ut id fiat? <strong>Theorema Weierstrassii<\/strong> pars erat desiderata in hoc aenigmate: cavet enim ut functio continua in intervallo clauso atque limitato definita non solum sit limitata, sed etiam valores suos extremos re vera attingat. In hac expositione enuntiationem eius recensimus, demonstrationem accuratam ex continuitate punctuali, compactione et axioma supremi structam elaboramus, deque interpretatione hodierna in terminis functionum continuarum in compactis disserimus. Propositum est ut, perlecto opere, non solam sententiam theorematum memineris, sed etiam intellegas cur verum sit et cur iterum atque iterum in analysi, in optimizatione et in exemplaribus adhibitis appareat.<\/em><\/p>\n<p style=\"text-align:center;\"><b>Proposita disciplinae<\/b><\/p>\n<ol>\n<li>\n    <strong>Enuntiationem Theorematis Weierstrassii comprehendere.<\/strong><br \/>\n    Praecisas conditiones theorematos agnoscere (functio continua in intervallo cluso et limitato <span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span>) atque conclusiones eius praecipuas: limitationem et existentiam valorum maximi et minimi.\n  <\/li>\n<li>\n    <strong>Theorema Weierstrassii secundum compactionem interpretari.<\/strong><br \/>\n    Propositum in lingua hodierna efformare: functiones continuae compacta in ea transformant ubi valores extremi attinguntur, coniungentes casum <span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span> cum ambitu generali analysis realis.\n  <\/li>\n<li>\n    <strong>Theorema Weierstrassii cum problematis optimizationis referre.<\/strong><br \/>\n    Agnoscere munus theorematos tamquam fundamentum theoreticum pro existentia maximorum et minimorum in multis problematis optimizationis unius variabilis, sive theoreticis sive adhibitis.\n  <\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>INDEX RERUM<\/u>:<\/b><br \/>\n<a href=\"#1\"><b>Introductio<\/b><\/a><br \/>\n<a href=\"#2\"><b>Enuntiatio Theorematis Weierstrassii<\/b><\/a><br \/>\n<a href=\"#3\">Demonstratio<\/a><br \/>\n<a href=\"#4\">Gradus 1: Continuitas punctalis in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/a><br \/>\n<a href=\"#5\">Gradus 2: Recubratio aperta cum continuitate coniuncta<\/a><br \/>\n<a href=\"#6\">Gradus 3: Compacitas <span dir=\"ltr\">[a,b]<\/span> et subrecubratio finita<\/a><br \/>\n<a href=\"#7\">Gradus 4: Constructio <span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span> cuius valor non pendet ab <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> (continuitas uniformis)<\/a><br \/>\n<a href=\"#8\">Gradus 5: Ex continuitate uniformi ad limitationem <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/a><br \/>\n<a href=\"#9\">Gradus 6: Existentia valorum maximi et minimi<\/a><br \/>\n<a href=\"#10\"><b>Interpretatio secundum compactionem et conclusio<\/b><\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/N5mSrhJgCds\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n<a name=\"1\"><\/a><\/br><\/p>\n<h2>Introductio<\/h2>\n<p>\n<strong>Theorema Weierstrassii de Valoribus Extremis<\/strong> est unum ex iis resultatibus quae, quamquam in primis unitatibus Analysis Realis apparere solent, revera magnam partem mathematicae applicatae tacite sustentant. Quoties in physica, oeconomia vel statistica de \u201cmaximanda\u201d vel \u201cminimanda\u201d quantitate sub certis condicionibus loquimur, re vera ideam admodum proximam ad illam huius theorematos adhibemus: functionem continuam in intervallo clauso ac limitato definitam <strong>non solum limitatam esse, sed etiam valores extremos suos re vera attingere<\/strong>.\n<\/p>\n<p>\nIntuitu videri potest \u201cmanifestum\u201d esse, si lineam continuam in segmento <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> depingimus, tum punctum aliquod altissimum atque aliud infimum exsistere debere. Nihilominus, satis est parvas mutationes in hypothesibus facere ut haec intuitio pessime deficiat: si intervallum aperimus, si functio desinit continua esse aut si dominium non est limitatum, maxima et minima simpliciter evanescere possunt. Theorema Weierstrassii ordinem huic intuitioni imponit et nobis accurate dicit <em>quando<\/em> in ea confidere possimus et <em>cur<\/em>.\n<\/p>\n<p>\nEx prospectu theoretico, hoc theorema primus est occursus gravis cum idea <strong>compactionis<\/strong>: lingua hodierna dictum est functionem continuam compacta in compacta transformare. Ex prospectu practico, hoc vertitur in existentiam solutionum pro multis problematis optimizationis unius dimensionis, atque erit pars fundamentalis pro resultatibus posterioribus sicut <b>Theorema de Valore Medio<\/b> et, ultimo, pro intellegendo lente Theorema Fundamentale Calculi.\n<\/p>\n<p>\nIn hoc capite enuntiabimus Theorema Weierstrassii atque demonstrationem eius diligenter exsequemur, nitentes in notione continuitatis in <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> et in axioma supremi. Propositum est ut hic textus tibi sit firmum subsidium: tum ad ipsum resultatim discendum, tum ad eum recognoscendum quoties opus sit illo uti ad alia theorematica probanda aut ad existentiam maximorum et minimorum in problematis concretis stricte demonstrandam.\n<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h2>Enuntiatio Theorematis Weierstrassii<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; background-color: #e0e0ff;\">\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=N5mSrhJgCds&amp;t=439s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Omnis functio <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> definita<\/span><\/strong><\/a> et continua in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b],<\/span><\/span> limitata est et valores minimum et maximum, <span class=\"katex-eq\" data-katex-display=\"false\">m<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">M<\/span>, habet, tales ut si <span class=\"katex-eq\" data-katex-display=\"false\">x\\in[a,b]<\/span>, tum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)\\in[m,M]<\/span><\/span>.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h3>Demonstratio<\/h3>\n<p>\nProbandum est, si <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f:[a,b]\\to\\mathbb{R}<\/span><\/span><\/strong> continua est in intervallo clauso ac limitato <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong>, tum <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> esse limitatam atque valorem maximum et minimum in <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> attingere. Demonstrationem in duas partes principales dividemus:\n<\/p>\n<ul>\n<li>Primum ostendemus continuitatem <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> in <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> efficere ut <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> <em>uniformiter continua<\/em> sit, atque hinc deducemus eam <strong>limitari<\/strong>.<\/li>\n<li>Deinde, utentes axioma supremi, probabimus <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> valores maximi et minimi in intervallo attingere.<\/li>\n<\/ul>\n<p><a name=\"4\"><\/a><\/br><\/p>\n<h4><b>Gradus 1:<\/b> Continuitas punctalis in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/h4>\n<p>\nEx hypothesi, <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> continua est in quoque puncto <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in[a,b]<\/span><\/span><\/strong>. Secundum definitionem continuitatis per <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span>, hoc significat:\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n(\\forall x_0\\in[a,b])(\\forall \\epsilon\\gt 0)(\\exists \\delta(x_0)\\gt 0)\n\n\\big(|x-x_0|\\lt\\delta(x_0)\\Rightarrow |f(x)-f(x_0)|\\lt\\epsilon\\big).\n\n<\/span>\n<\/p>\n<p>\nHoc loco numerus <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta(x_0)<\/span><\/span><\/strong> a puncto <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span><\/strong> pendere potest. Propositum proximum est, ex his <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta(x_0)<\/span><\/span><\/strong> unum numerum <strong><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/strong> construere qui ab <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span><\/strong> non pendeat et qui omnibus punctis intervalli simul valeat.\n<\/p>\n<p><a name=\"5\"><\/a><\/br><\/p>\n<h4><b>Gradus 2:<\/b> Recubratio aperta cum continuitate coniuncta<\/h4>\n<p>\nFigeamus <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon\\gt 0<\/span><\/span><\/strong> quendam. Pro quoque <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in[a,b]<\/span><\/span><\/strong>, continuitas <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> sinit nos eligere numerum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta(x_0)\\gt 0<\/span><\/span><\/strong> talem ut\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|x-x_0|\\lt\\delta(x_0)\\Rightarrow |f(x)-f(x_0)|\\lt\\frac{\\epsilon}{2}.\n\n<\/span>\n<\/p>\n<p>\nEx his valoribus definimus, pro quoque <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in[a,b]<\/span><\/span><\/strong>, intervallum apertum\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\nI_{x_0}=\\left(x_0-\\frac{\\delta(x_0)}{2},\\,x_0+\\frac{\\delta(x_0)}{2}\\right).\n\n<\/span>\n<\/p>\n<p>\nQuodque <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{x_0}<\/span><\/span><\/strong> est apertum in <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span>, praeterea familia\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n\\{I_{x_0}\\}_{x_0\\in[a,b]}\n\n<\/span>\n<\/p>\n<p>\nconstituit <strong>recubratio apertam<\/strong> <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong>. Re vera, dato puncto quolibet <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y\\in[a,b]<\/span><\/span><\/strong>, satis est sumere <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0=y<\/span><\/span><\/strong>; ex constructione, <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y\\in I_y<\/span><\/span><\/strong>. Ita quodque punctum intervalli saltem uni ex apertis <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{x_0}<\/span><\/span><\/strong> inest.\n<\/p>\n<p>\nHaec familia apertorum est, generatim, <strong>infinita<\/strong> (est enim unum pro quoque <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in[a,b]<\/span><\/span><\/strong>). Hic locus est ubi compacitas <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> munus suum agit.\n<\/p>\n<p><a name=\"6\"><\/a><\/br><\/p>\n<h4><b>Gradus 3:<\/b> Compacitas <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> et subrecubratio finita<\/h4>\n<p>\nEx Theoremate Heine\u2013Borel scimus subensemble <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span><\/span> compactum esse si et solum si clausum et limitatum est. Intervallum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> clausum est et limitatum, igitur compactum est. Ex definitione compactionis hoc significat:\n<\/p>\n<p>\nEx <strong>omni<\/strong> recubratione aperta <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> (etiam si infinita sit) extrahi posse <strong>subrecubrationem finitam<\/strong>.\n<\/p>\n<p>\nHanc proprietatem ad recubrationem apertam <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{I_{x_0}\\}_{x_0\\in[a,b]}<\/span><\/span><\/strong> applicantes, sequitur exsistere puncta <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1,\\dots,x_N\\in[a,b]<\/span><\/span><\/strong> talia ut intervalla eis respondentia sint\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\nI_{x_1},\\, I_{x_2},\\,\\dots,\\,I_{x_N}\n\n<\/span>\n<\/p>\n<p>\nadhuc totum intervallum recubrant:\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n[a,b]\\subset I_{x_1}\\cup I_{x_2}\\cup\\cdots\\cup I_{x_N}.\n\n<\/span>\n<\/p>\n<p>\nIta transivimus ab familia infinita intervallorum apertorum ad subrecubrationem cum solis <strong>numeris finitis<\/strong> intervallorum, sine amissione proprietatis recubrendi <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong>.\n<\/p>\n<p><a name=\"7\"><\/a><\/br><\/p>\n<h4><b>Gradus 4:<\/b> Constructio <span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span> quod ab <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> non pendet (continuitas uniformis)<\/h4>\n<p>\nEx subrecubratione finita definimus numerum\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n\\delta=\\min\\left\\{\\frac{\\delta(x_1)}{2},\\frac{\\delta(x_2)}{2},\\dots,\\frac{\\delta(x_N)}{2}\\right\\}.\n\n<\/span>\n<\/p>\n<p>\nCum sit minimum quantitatis finitae numerorum positivorum, constat <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta\\gt 0<\/span><\/span><\/strong>. Ostendemus hoc <strong><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/strong> valere pro <strong>omni<\/strong> puncto <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in[a,b]<\/span><\/span><\/strong>, id est, non pendere ab electione <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span><\/strong>.\n<\/p>\n<p>\nNunc sumamus:\n<\/p>\n<ul>\n<li>punctum arbitrarium <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in[a,b]<\/span><\/span><\/strong>, et<\/li>\n<li>punctum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in[a,b]<\/span><\/span><\/strong> tale ut <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|x-x_0|\\lt\\delta<\/span><\/span><\/strong>.<\/li>\n<\/ul>\n<p>\nQuoniam intervalla <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{x_1},\\dots,I_{x_N}<\/span><\/span><\/strong> <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> recubrant, punctum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span><\/strong> saltem uni eorum inest, dicamus <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{x_j}<\/span><\/span><\/strong> pro quodam <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">j\\in\\{1,\\dots,N\\}<\/span><\/span><\/strong>. Ex definitione <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{x_j}<\/span><\/span><\/strong>, hoc significat\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|x_0-x_j|\\lt\\frac{\\delta(x_j)}{2}.\n\n<\/span>\n<\/p>\n<p>\nPraeterea, ex definitione <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span><\/strong> habemus <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta\\le\\frac{\\delta(x_j)}{2}<\/span><\/span><\/strong>, unde ex <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|x-x_0|\\lt\\delta<\/span><\/span><\/strong> sequitur\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|x-x_0|\\lt\\frac{\\delta(x_j)}{2}.\n\n<\/span>\n<\/p>\n<p>\nAdhibita inaequalitate trianguli,\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|x-x_j|\\le |x-x_0|+|x_0-x_j|\n\n\\lt \\frac{\\delta(x_j)}{2}+\\frac{\\delta(x_j)}{2}\n\n=\\delta(x_j).\n\n<\/span>\n<\/p>\n<p>\nEx electione <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta(x_j)<\/span><\/span><\/strong> (continuitatis <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> in <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_j<\/span><\/span><\/strong> pro valore <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon\/2<\/span><\/span><\/strong>), inaequalitates <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|x_0-x_j|\\lt\\delta(x_j)<\/span><\/span><\/strong> et <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|x-x_j|\\lt\\delta(x_j)<\/span><\/span><\/strong> implicant\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|f(x_0)-f(x_j)|\\lt\\frac{\\epsilon}{2}\n\n\\quad\\text{et}\\quad\n\n|f(x)-f(x_j)|\\lt\\frac{\\epsilon}{2}.\n\n<\/span>\n<\/p>\n<p>\nIterum adhibita inaequalitate trianguli obtinetur\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|f(x)-f(x_0)|\n\n\\le |f(x)-f(x_j)| + |f(x_j)-f(x_0)|\n\n\\lt \\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}\n\n=\\epsilon.\n\n<\/span>\n<\/p>\n<p>\nCum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span><\/strong> et <strong><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/strong> fuerint arbitraria, demonstravimus pro illo <strong><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/strong> initio fixo exsistere <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta\\gt 0<\/span><\/span><\/strong> ab <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span><\/strong> independente tale ut\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n(\\forall x_0\\in[a,b])(\\forall x\\in[a,b])\n\n\\big(|x-x_0|\\lt\\delta\\Rightarrow |f(x)-f(x_0)|\\lt\\epsilon\\big).\n\n<\/span>\n<\/p>\n<p>\nSi <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span><\/strong> sub nomine <strong><span class=\"katex-eq\" data-katex-display=\"false\">y<\/span><\/strong> denotemus, hoc scribitur:\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n(\\forall \\epsilon\\gt 0)(\\exists \\delta\\gt 0)(\\forall x,y\\in[a,b])\n\n\\big(|x-y|\\lt\\delta\\Rightarrow |f(x)-f(y)|\\lt\\epsilon\\big),\n\n<\/span>\n<\/p>\n<p>\nquae est ipsa definitio <strong>continuitatis uniformis<\/strong> <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> in <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong>. In sequentibus hoc tantum pro casu <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon=1<\/span><\/span><\/strong> adhibebimus.\n<\/p>\n<p><a name=\"8\"><\/a><\/br><\/p>\n<h4><b>Gradus 5:<\/b> Ex continuitate uniformi ad limitationem <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/h4>\n<p>\nAdhibeamus nunc continuitatem uniformem cum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon=1<\/span><\/span><\/strong>. Exstat numerus <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta_1\\gt 0<\/span><\/span><\/strong> talis ut pro omnibus <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in[a,b]<\/span><\/span><\/strong> valeat\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|x-y|\\lt\\delta_1\\Rightarrow |f(x)-f(y)|\\lt 1.\n\n<\/span>\n<\/p>\n<p>\nIam dividimus intervallum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> in numerum finitum subintervallorum quarum longitudo sit minor quam <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta_1<\/span><\/span><\/strong>. Id est, eligimus integrum <strong><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/strong> et puncta\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\na = x_0 \\lt x_1 \\lt \\cdots \\lt x_n = b\n\n<\/span>\n<\/p>\n<p>\nita ut pro singulis <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0,1,\\dots,n-1<\/span><\/span><\/strong> valeat\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\nx_{k+1}-x_k\\lt\\delta_1.\n\n<\/span>\n<\/p>\n<p>\nConsideremus nunc finitum valorem collectum\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n\\{f(x_0),f(x_1),\\dots,f(x_{n-1})\\}.\n\n<\/span>\n<\/p>\n<p>\nCum sit numerorum realium collectio finita, definire possumus\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\nC = \\max\\{|f(x_k)| \\;|\\; k=0,1,\\dots,n-1\\}.\n\n<\/span>\n<\/p>\n<p>\nOstendemus <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">C+1<\/span><\/span><\/strong> esse cotam superiorem absoluto valore <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> in toto intervallo <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong>. Sit <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in[a,b]<\/span><\/span><\/strong> punctum arbitrarium. Tum exstat index <strong><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/strong> talis ut <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in[x_k,x_{k+1}]<\/span><\/span><\/strong>. In specie,\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|x-x_k|\\le x_{k+1}-x_k\\lt\\delta_1.\n\n<\/span>\n<\/p>\n<p>\nEx continuitate uniformi cum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon=1<\/span><\/span><\/strong>, ex <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|x-x_k|\\lt\\delta_1<\/span><\/span><\/strong> sequitur\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|f(x)-f(x_k)|\\lt 1.\n\n<\/span>\n<\/p>\n<p>\nInequalitate trianguli adhibita:\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|f(x)|\\le |f(x)-f(x_k)| + |f(x_k)| \\lt 1 + |f(x_k)| \\le 1 + C.\n\n<\/span>\n<\/p>\n<p>\nCum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in[a,b]<\/span><\/span><\/strong> fuerit arbitrarium, concludimus\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n|f(x)|\\le C+1 \\quad \\text{pro omnibus } x\\in[a,b],\n\n<\/span>\n<\/p>\n<p>\nid est, functio <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> est <strong>limitata<\/strong> in <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong>.\n<\/p>\n<p><a name=\"9\"><\/a><\/br><\/p>\n<h4><b>Gradus 6:<\/b> Existentia valorum maximi et minimi<\/h4>\n<p>\nDefinimus collectum valorum quos functio in intervallo sumit:\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\nH=\\{f(x)\\;|\\;x\\in[a,b]\\}\\subset\\mathbb{R}.\n\n<\/span>\n<\/p>\n<p>\nIam scimus <strong><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/strong> non vacuum esse (cum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> non vacuum sit) atque limitatum, ita per axioma supremi exsistunt numeri reales\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\nM=\\sup H,\\qquad m=\\inf H.\n\n<\/span>\n<\/p>\n<p>\nProbandum est <strong><span class=\"katex-eq\" data-katex-display=\"false\">M<\/span><\/strong> attingi ut valor functionis, id est, exsistere <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1\\in[a,b]<\/span><\/span><\/strong> cum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x_1)=M<\/span><\/span><\/strong>. Procedemus per reductionem ad absurdum.\n<\/p>\n<p>\nPonamus <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span><\/strong> numquam valorem <strong><span class=\"katex-eq\" data-katex-display=\"false\">M<\/span><\/strong> attingere, id est:\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n(\\forall x\\in[a,b])\\big(f(x)\\lt M\\big).\n\n<\/span>\n<\/p>\n<p>\nHac sub hypothesi, functio\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\ng(x)=\\frac{1}{M-f(x)}\n\n<\/span>\n<\/p>\n<p>\nbene definita est et positiva pro omni <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in[a,b]<\/span><\/span><\/strong>, cum ex hypothesi <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M-f(x)\\gt 0<\/span><\/span><\/strong>. Praeterea, cum <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> continua sit et <strong><span class=\"katex-eq\" data-katex-display=\"false\">M<\/span><\/strong> constans sit, etiam <strong><span class=\"katex-eq\" data-katex-display=\"false\">g<\/span><\/strong> continua est. Per primam demonstrationis partem, omnis functio continua in <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> limitata est, unde exstat numerus <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N\\gt 0<\/span><\/span><\/strong> talis ut\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n(\\forall x\\in[a,b])\\big(g(x)\\le N\\big).\n\n<\/span>\n<\/p>\n<p>\nIn specie, pro omni <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in[a,b]<\/span><\/span><\/strong> verificatur\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n\\frac{1}{M-f(x)} = g(x)\\le N,\n\n<\/span>\n<\/p>\n<p>\nquod aequivalet\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\nM-f(x)\\ge \\frac{1}{N}\n\n\\quad\\Rightarrow\\quad\n\nf(x)\\le M-\\frac{1}{N}.\n\n<\/span>\n<\/p>\n<p>\nHoc significat omnes valores <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span><\/strong> in <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> minores vel aequales esse <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M-\\frac{1}{N}<\/span><\/span><\/strong>. In specie, supremum <strong><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/strong> satisfacit\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\n\\sup H\\le M-\\frac{1}{N}\\lt M,\n\n<\/span>\n<\/p>\n<p>\nquod pugnat cum definitione <strong><span class=\"katex-eq\" data-katex-display=\"false\">M<\/span><\/strong> tamquam supremi <strong><span class=\"katex-eq\" data-katex-display=\"false\">H<\/span><\/strong>. Ergo falsum erat principium nostrum, atque exsistere debet punctum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_1\\in[a,b]<\/span><\/span><\/strong> tale ut\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\nf(x_1)=M.\n\n<\/span>\n<\/p>\n<p>\nRatio omnino similis, ad infimum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m=\\inf H<\/span><\/span><\/strong> applicata (exempli gratia considerantes functionem <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h(x)=-f(x)<\/span><\/span><\/strong>), demonstrat exsistere punctum <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_2\\in[a,b]<\/span><\/span><\/strong> tale ut\n<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\n\nf(x_2)=m.\n\n<\/span>\n<\/p>\n<p><a name=\"10\"><\/a><\/br><\/p>\n<h2>Interpretatio secundum compactionem et conclusio<\/h2>\n<p>\nProbavimus omnem functionem continuam <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f:[a,b]\\to\\mathbb{R}<\/span><\/span><\/strong> limitatam esse atque valores maximi et minimi in <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> attingere. In lingua moderna analysis, hoc ita explicatur: in <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span><\/span><\/strong>, intervalla clausa et limitata sicut <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span><\/strong> sunt compacta, et functiones continuae compacta in compacta transferunt.\n<\/p>\n<p>\nIn specie, si <strong><span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/strong> compactum est et <strong><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/strong> continua est in <strong><span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/strong>, tum imago <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(I)<\/span><\/span><\/strong> est subcollectio compacta <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span><\/span><\/strong>. Hoc spondet <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(I)<\/span><\/span><\/strong> esse limitatum et in eo valores maximum et minimum re vera attingi, quod est ipsum quod Theorema Weierstrassii continet.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Theorema Weierstrassii de Valoribus Extremis Cur in tot problematis optimizationis fere pro certo habetur \u201cmaximum exsistere\u201d aut \u201csemper minimum dari\u201d in quodam intervallo, cum re vera nihil cogat ut id fiat? Theorema Weierstrassii pars erat desiderata in hoc aenigmate: cavet enim ut functio continua in intervallo clauso atque limitato definita non solum sit limitata, sed [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35255,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":5,"footnotes":""},"categories":[1328,1298],"tags":[],"class_list":["post-35287","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculus-differentialis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>THEOREMA WEIERSTRASSII DE VALORIBUS EXTREMIS - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Intellegere Theorema Weierstrassii ab initio: continuitas, compacitas et demonstratio gradatim explicata ad problemata maximorum et minimorum applicata.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"THEOREMA WEIERSTRASSII DE VALORIBUS EXTREMIS\" \/>\n<meta property=\"og:description\" content=\"Intellegere Theorema Weierstrassii ab initio: continuitas, compacitas et demonstratio gradatim explicata ad problemata maximorum et minimorum applicata.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2024-12-20T13:00:16+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-12-11T17:08:40+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Weierstrass-1-1024x683.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"THEOREMA WEIERSTRASSII DE VALORIBUS EXTREMIS\" \/>\n<meta name=\"twitter:description\" content=\"Intellegere Theorema Weierstrassii ab initio: continuitas, compacitas et demonstratio gradatim explicata ad problemata maximorum et minimorum applicata.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Weierstrass-1.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"8 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"THEOREMA WEIERSTRASSII DE VALORIBUS EXTREMIS\",\"datePublished\":\"2024-12-20T13:00:16+00:00\",\"dateModified\":\"2025-12-11T17:08:40+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/\"},\"wordCount\":2302,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Weierstrass-1.jpg\",\"articleSection\":[\"Calculus Differentialis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/\",\"url\":\"https:\/\/toposuranos.com\/material\/la\/theorema-weierstrassii-de-valoribus-extremis\/\",\"name\":\"THEOREMA WEIERSTRASSII DE VALORIBUS EXTREMIS - 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