Differentiability at 0 I am having a problem with this exercise. Please help.

Let $\alpha >1$. Show that if $|f(x)| \leq |x|^\alpha$, then $f$ is differentiable at $0$.


Use the definition of the derivative. It is clear that $f(0)=0$. Note that if $h\ne 0$ then
$$\left|\frac{f(h)-0}{h}\right| \le |h|^{\alpha-1}.$$
Since $\alpha\gt 1$, $|h|^{\alpha-1}\to 0$ as $h\to 0$.
