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.
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:
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.
\[Cf, \text{ where $C$ is a constant,}\] \[\text{and }f\big(g(x) \big) \text{: the composite function $f$ of $g$ of $x$}\]
The following Example illustrates the type of question you might be asked about these ideas.
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:
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!
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