Summary:
This class addresses the relationship between the limit and the continuity of a function, starting with an intuitive and formal explanation of the term. Continuity at a point and in a set is explored, detailing the necessary conditions for a function to be continuous. The algebraic properties of continuous functions are also presented, including sums, products, quotients, and function composition. Finally, practical examples are solved to analyze the continuity of different functions and calculate limits, relating them to the concept of continuity.

Learning Objectives:
At the end of this class, the student will be able to:

1. Analyze the continuity of a function at a point by verifying the defined conditions.
2. Evaluate whether a function is continuous on a given set using the formal definition of continuity.
3. Demonstrate the continuity of a function based on its algebraic properties.
4. Classify points of discontinuity into different types according to the conditions that are not met.

CONTENT INDEX:
Introduction
Continuity at a Point
Continuity in a Set
Properties of Continuous Functions
Exercises

Introduction

Intuitively speaking, continuity of a function is understood as the property that allows its graph to be drawn “without lifting the pencil.” This notion, which is quite simple to understand on its own, is not sufficient, however, if mathematical rigor is desired. A minimum of formalities is required. Formally speaking, the continuity of functions is closely linked to the concept of limit, and this is what we will study in detail.

Continuity at a Point

A function f(x) is continuous at a point x=x_0 if and only if the following conditions are met:

• f(x) is defined at x_0
• The limit \displaystyle \lim_{x\to x_0} f(x) exists
• \displaystyle \lim_{x\to x_0} f(x) = f(x_0)

All of this can be mathematically synthesized through the expression

\left(\forall \epsilon \gt 0 \right) \left(\exists \delta \gt 0 \right) \left(0\lt |x-x_0|\lt \delta \rightarrow |f(x) - f(x_0)|\lt \epsilon\right)

This is precisely the mathematical definition of a limit, where the value “L” has been replaced by “f(x_0)”

Continuity in a Set

The definition of pointwise continuity can be extended to continuity for all points within a certain set. We say that f(x) is continuous within a set I\subseteq Dom(f) if it is continuous at all points x_0\in I. This can be mathematically synthesized through the following expression.

\left(\forall x_0 \in I\subseteq Dom(f) \right)\left(\forall \epsilon \gt 0 \right) \left(\exists \delta \gt 0 \right) \left(0\lt |x-x_0|\lt \delta \rightarrow |f(x) - f(x_0)|\lt \epsilon\right)

Properties of Continuous Functions

Algebra of Continuous Functions

If f and g are continuous functions on I\subseteq Dom(f)\cap Dom(g) then we will have:

• f\pm g is continuous on I
• f\cdot g is continuous on I
• If \alpha\in\mathbb{R}, then \alpha f is continuous on I
• f/g is continuous on I as long as g\neq 0
• If \alpha,\beta\in\mathbb{R} and \beta\neq 0, then f^{\alpha/\beta} is continuous on I, as long as f^{\alpha/\beta} is well defined on I.

All these properties, which seem difficult to prove from the formal definition of continuity, are actually very easy because they are practically analogous to the algebra of limits that we have already proven, so we are exempt from doing them.

Composition of Continuous Functions

If f is continuous on I\subseteq Dom(f) and g is continuous on f(I), then the composition (g\circ f)(x) = g(f(x)) is continuous on I. This, as is already inferred, is a direct consequence of the laws of composition for the limits of functions.

Exercises

1. Analyze the continuity of the following functions
   

   a.	y=\dfrac{1}{x-2} - 3x	SOLUTION
   b.	y=\dfrac{x+1}{x^2-4x+3}	SOLUTION
   c.	y=|x-1| + \sin(x)	SOLUTION
   d.	y=\dfrac{\cos(x)}{x}	SOLUTION
   e.	y=csc(2x)	SOLUTION
   f.	y=\dfrac{x\tan(x)}{x^2 + 1}	SOLUTION
   g.	y=\sqrt{2x + 3}	SOLUTION
   h.	y=\dfrac{x+2}{\cos(x)}	SOLUTION
   i.	y=\dfrac{\sqrt{x^4 + 1}}{1+\sin^2(x)}	SOLUTION

2. Calculate the following limits. Are the functions continuous at the points they approach?
   

   a.	\displaystyle\lim_{x\to\pi} \sin(x -\sin(x))	SOLUTION
   b.	\displaystyle\lim_{x\to 0} \sin\left(\dfrac{\pi}{2}\cos(\tan(x)) \right)	SOLUTION
   c.	\displaystyle\lim_{x\to 1} \sec\left(x\sec^2(x) - \tan^2(x) -1 \right)	SOLUTION
   d.	\displaystyle\lim_{x\to 0} \tan\left(\dfrac{\pi}{4}\cos\left(\sin\left(x^{1/3}\right)\right) \right)	SOLUTION
   e.	\displaystyle\lim_{x\to 0} \cos\left(\dfrac{\pi}{\sqrt{19 - 3\sec{2x}}} \right)	SOLUTION
   f.	\displaystyle\lim_{x\to \pi/6} \sqrt{\csc^2(x) + 5\sqrt{3}\tan(x)}	SOLUTION