Introduction

Before getting started with the actual content, it is important to know that this article contains one topic, but after you finish this article and the next article, you will be ready for unit 1.13. You are already familiar with the different types of discontinuities, thus we can dive into continuity.

[Editor's note: this topic is SPLIT between 1.1 1 and 1.12]

There is a definition for continuity that a function must follow in order to determine if it is continuous or not at a point: A function $f(x)$ is continuous at $x=c$ when $f(c)$ exists, $\underset{x\to c^{-}}{\lim }f(x)=f(c)$, and $\underset{x\to c^{+}}{\lim }f(x)=f(c)$. In short, the definition is that the left and right-sided limits of the function at a point must equal the value of the function at that point.

Now, we can see that the discontinuities we explored in the previous article violate these conditions. Here is an example that shows the function being continuous at a point.

Example 1:

Show that $f\left(x\right)=6x^7-41x^2+21$ is continuous at $x=2$. 

$f\left(2\right)=6{\left(2\right)}^7-41{\left(2\right)}^2+21=625$,  \displaystyle \lim_{x \to 2^-} f(x) =  \displaystyle \lim_{x \to 2^+} f(x) = f\left(2\right) = 625$.

Therefore, we have proved that $f(x)$ is continuous at $x=2$.

Knowing a function is continuous also allows for us to find values of functions.

Practice

Here is one practice question to see how much you’ve learned.