Have to grade a midterm where one of the true/false questions boils down to whether or not $f(x)=(x^2)^x$ is differentiable at 0. I'm not sure of the answer.
For one thing, the continuity of $f(x)$ is author-dependent since it hinges on what $0^0$ is taken to be. Let's assume $0^0$ is defined as 1, so that $f(x)$ is continuous at $0$.
For all $x\not=0$, $f'(x)=(\ln(x^2)+2)(x^2)^x$. Thus, $\lim_{x\to 0}f'(x)=-\infty$. Can we somehow deduce that $f'(0)$ is nonexistent from this, e.g., some kind of result of the form "wherever f is differentiable, it is continuously differentiable"?