Everyone studying mathematics needs to know some typical functions and some important characteristics. At the lowest level you need to know the trig functions and their (as applicable) domain, asymptotes, midlines, periods, etc and for exponentials and logarithms you should know the domains and asymptotes. At a slightly more advanced level, students should be aware of the function $latex \frac{\sin(x)}{x}$ and how its limit exists at 0 (and is equal to 1) even though the function isn't defined there. Or that $latex x\sin(1/x)$ is a function whose limit at 0 exists and can be determined using the Squeeze Theorem. The graph show above is also a function that needs to filed away but, surprisingly, it isn't well known. I say that because I have trouble finding it in a typical calculus text (or the lesson it teaches). That's a shame; the function is $latex f(x)=x^2\sin(1/x^2)$ if $latex x \neq 0$ and $latex f(0)=0$.

Take the derivative and you'll get $latex f'(x)=x\sin(1/x^2)-(2/x)\cos(1/x^2)$. And the derivative at 0? Chances are overwhelming that your students will say the derivative doesn't exist at 0 because $latex f'(0)$ isn't defined. But that's wrong. That won't be apparent by looking at the graph of the derivative:For the typical students who dive for a calculator so they can avoid thinking for themselves, they'll get to see that the calculator isn't going to help out here. They'll have to use the definition of the derivative (and the Squeeze Theorem). That's good practice. Ultimately this function shows that derivative can exist at a point even if the formula $f'(x)$ doesn't exist there. For all these reasons, this graph should appear in every calculus book.

The PDFs have been posted on the Graphics page.