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.
So for a function to be continuous at a number, all three statements must hold. Said differently:
Here are three examples showing how a discontinuity might arise. The graphs of these functions show the discontinuity if you look close enough.
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.
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.
Here are a few basic examples, on how you can use the previous theorems to construct continuous functions from basic 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.
Here is another example demonstrating finding parameters to obtain a continuous function.
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.
And here are some examples using this theorem.
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.
Intermediate Value Theorem
The Intermediate Value Theorem is extremely useful.
Exercises on Continuous Functions
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!