I need to find an example of a function $f:I$ to $R$ such that $f$ is uniformly continuous $f'$ exists but $f'$ is not bounded? I'm fairly stuck with this - it's part of a review and I guess I never handled any of these types of problems very well. Any direction is appreciated.
(This is a standard example, it has the advantage that the issue is not an "infinite derivative".)
Take $f(x)=x^2\sin(1/x^2)$ for $x\ne0$ and $f(0)=0$. This function is continuous and therefore uniformly continuous on any bounded interval, for example $I=[0,1]$. On the other hand, $f'(0)=0$ and $f'(x)=2x\sin(1/x^2)-(2/x)\cos(1/x^2)$ for $x\ne0$, which is unbounded on any neighborhood of $0$.
If you want an example on unbounded intervals, pick $x_0>0$ with $f'(x_0)=0$, start with the $f$ from the previous paragraph, but only on $[-x_0,x_0]$ and extend it to $\mathbb R$ by setting $f(t)=f(x_0)$ for $|t|>x_0$.