Matheno - Learn Well and Excel

D.3 Continuous functions; Continuity theorems

To conclude our discussion of continuity, let’s define continuous functions, and introduce some continuity theorems. This screen will be quick: we’re introducing ideas that we’ll need later, but there isn’t a whole lot of practice worth doing now. That said, the concepts themselves are crucial to have in mind.

Continuous functions

The definition of a continuous function is no surprise: a function is a continuous function if it’s continuous at every number a in its domain.

And we state without proof that the following functions are continuous everywhere they are defined:

Continuous functions

  • polynomials
  • trig functions
  • exponentials
  • root functions (e.g., $\sqrt{x}$)

  • rational functions (quotient of two polynomials)
  • inverse trig functions
  • logarithmic functions

 

Do not work to memorize this list
There’s no need to memorize this list; instead, just remember that the functions you normally think of are continuous where they are defined. (We emphasize that last phrase because, as we saw on the preceding screen, a function like $f(x) = \dfrac{1}{x-1}$ is of course not continuous at $x=1,$ but is continuous everywhere else.) Note that proving that these functions are all continuous is well beyond the scope of an introductory Calculus course, and we point you once again to an upper division Mathematics course in Analysis if you’re interested in such things!

Continuity theorems

Let’s also state theorems that codify how continuous functions can be put together to make other continuous functions. Essentially, this all works the way you think it should: adding two continuous functions together gives you a new continuous function; multiplying a continuous function by a constant gives you another new continuous function; dividing one continuous function by another gives you yet another continuous function (everywhere the function in the denominator is defined); and so forth.

Theorems regarding combining continuous functions
If f and g are continuous functions at $x=a,$ then so are

\begin{align*}
&f+g, \\[8px] &f-g,
\end{align*}
\begin{align*}
&fg, \\[8px] &\frac{f}{g}, \text{ where }g(a)\ne 0,
\end{align*}

\[Cf, \text{ where $C$ is a constant,}\] \[\text{and }f\big(g(x) \big) \text{: the composite function $f$ of $g$ of $x$}\]

 
The following Example illustrates the type of question you might be asked about these ideas.

Example 1: Continuity of $f(x) = \sqrt[3]{\dfrac{\cos(x)}{x^2 + 5}}$

Determine where the function $f(x) = \sqrt[3]{\dfrac{\cos(x)}{x^2 + 5}}$ is continuous.

Solution.
The polynomial $x^2 + 5$ is continuous for all real values of x, $(-\infty, \infty)$. Since $x^2 + 5$ never equals zero, we don’t have to worry about the denominator of f equaling zero. ($x^2 + 5$ never equals zero because $x^2$ is never less than 0, so $x^2 + 5$ is never less than 5.)

The function $\cos (x)$ is defined and continuous for $(-\infty, \infty).$

And the cube-root function $\sqrt[3]{\phantom{x}}$ can take on any input value, positive, negative or zero. It is continuous on $(-\infty, \infty).$

Hence f is continuous on $(-\infty, \infty) \; \cmark$.

The list of continuous functions above, plus these theorems, make it easy to state that a particular function is continuous on an interval of interest. And a function’s continuity is a prerequisite to being able to apply other theorems to and draw other conclusions about the function — as we’ll see starting on the next screen. For now, let’s remind you again that we used to say a function had to “be nice and smooth, with no sudden jumps or gaps,” whereas now we can simply say that it is continuous.

With that, here are a few problems to put your new knowledge to use:

Practice Problems: Continuous Functions

This content is available to logged-in users.
To interact with all of the content of this page, including Check Questions and Practice Problems (each with a complete solution), you must be logged in.
  • You can immediately sign in for free using your Facebook, Google, or Apple account.
  • If you would rather use your dedicated Matheno.com account, please enter your username and password below. (Don't have a free account but would like one? You can create it in 60 seconds here.)
This content is available to logged-in users.

On the next screen we conclude our study of limits and continuity by introducing the Intermediate Value Theorem.

For now, what questions or thoughts do you have about continuous functions, or continuity in general? Head on over to the Forum and join the discussion!

The Upshot

  1. The functions in the box at the top of this screen are all continuous functions where they are defined. They include all of the ones you know well: polynomials, root functions, trig functions, exponentials, logarithms, and so forth.
  2. Functions made by combining those functions (adding them, multiplying them, composing one with another, …) are also continuous.