Continuous Functions (Meaning and Examples)

by Dave
(DAVE)—

Continuity is one of the essential concepts in all of the calculus; indeed, all of mathematics. So in this article, I discuss continuous functions and the cases where a discontinuity may occur. I also present one-sided continuity and the removability of a continuity. In other words, how to extend a function so that it has no discontinuities.

We discuss continuous functions, one-sided and two-sided continuity, and removable continuity. The infamous Intermediate Value Theorem is considered at the end. First you need a good understanding of limits of functions.

A function is called continuous whenever sufficiently small changes in the input results in arbitrarily small changes in the output.

definition of a continuous function at a number

So for a function to be continuous at a number, all three statements must hold. Said differently:

continuous function at a number
discontinuous meaning and continuity on an interval

Discontinuity

Here are three examples showing how a discontinuity might arise. The graphs of these functions show the discontinuity if you look close enough.

three examples on how a discontinuity might arise
three graphs showing how a discontinuity might arise

Continuous Functions

The next theorem says that continuity on functions operations are compatible. The sum, difference, product, quotient, and composition of continuous functions are continuous functions, provided the function is defined.

continuous functions are compatible with function operations

As an extension of the previous theorem we see that polynomial, rational, trigonometric functions, as well as, the inverse of a continuous function is continuous where it is defined.

basic types of continuous functions

Here are a few basic examples, on how you can use the previous theorems to construct continuous functions from basic continuous functions.

some examples of continuous functions

Determining Parameters for Continuity

In this following example, we are given a piecewise function consists of a piece of parabolas and are ask to find parameters so we can piece them together, obtaining a continuous function.

example determining continuity by finding parameters
a continuous piecewise function

Here is another example demonstrating finding parameters to obtain a continuous function.

finding parameters so that a piecewise defined function is continuous

The Limit of a Composition

The next theorems guarantees that the limit of a composition of function is found by evaluating the function after taking the limit of the other.

composition limit theorem

And here are some examples using this theorem.

using the composition limit theorem

One-Sided Continuity

A function is continuous from the right at a number if and only if the right-sided limit at this number is the same as the function evaluated at this number. For example, the square root function is continuous from the right at 0.

continuous from the left and continuous from the right
example of continuous from the right at

Intermediate Value Theorem

The Intermediate Value Theorem is extremely useful.

the statement of the Intermediate Value Theorem
example using the Intermediate Value Theorem

Exercises on Continuous Functions

In the video, I work through the details of the next three exercises.

sketch the graph of a piecewise function and find some limits and determine continuity
find constant so that a function will be a continuous function

If you like, I can make another video with the solutions to the following exercises. Just leave a comment.

for each of the following functions determine the largest set on which the function will be continuous
fix the removable discontinuity

Calculus 1 (Explore, Discover, Learn) Series

If you would like me to make a video with the solutions to some of the exercises let me know in the comments.

This article (and accompanying video) are a part of a series of articles (and videos) called the Calculus 1 (Explore, Discover, Learn) Series. Also, I put together for you a getting started with calculus 1 page and a video playlist for calculus one.

Have fun in your studies!