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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 $0$.

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    What you have is not *an expression similar to the limit* but a precise definition involving a quite definite limit.2012-10-17

1 Answers 1

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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 $h\to 0$.