Introduction

Before getting started with the actual content, it is important to know that this article contains one topic, but after you finish the previous article and this 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.

Example 1:

Suppose $f(x)$ is continuous for all values of $x$, find the value of k.

$f(x)=\{\begin{array}{ll}kx+6& \text{if }x<-2,\\ x^2-2x& \text{if }x\ge 2.\end{array}$

Since the function is continuous everywhere, it must be continuous at $x=2$. Thus, we can find $k$ using the definition of continuity. 

Taking the left and right side limits of the function yields…

$\underset{x\to 2^{-}}{\lim }f(x)=2k+6$ and $\underset{x\to 2^{+}}{\lim }f(x)=0$.

We also know that $f(2)=0$, so using the definition, we get  $\underset{x\to 2^{-}}{\lim }f(x)=\underset{x\to 2^{+}}{\lim }f(x)=f(2)$. Now, plugging in the values we found earlier yields  $2k+6=0=0$. So, $k=-3$.

Make sure to read the question, and if it asks for a value like $f(k)$ make sure to plug $k$ back in and don’t stop after finding out the value.

Finally, a function is continuous on an interval if it is continuous on each point in its interval. Many functions are continuous over their domains. The ones listed on the College Board course and exam description include: polynomial, rational, power, exponential, logarithmic, and trigonometric functions.

Practice

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