Let f:RR be a function such that |f(x)| x^2 for all xR.

Question

Let $f:\mathbb{R}	o\mathbb{R}$ be a function such that $|f(x)|\le x^2$ for all $x\in\mathbb{R}$. Then, at $x=0$, $f$ is:

Accepted Answer

$|f(0)|\le0\Rightarrow f(0)=0$. $\left|\dfrac{f(h)}{h}ight|\le|h|	o0$, so by the Sandwich theorem $f'(0)=0$ exists, i.e. $f$ is differentiable at $0$, and hence also continuous there.

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