Show that a function bounded by $Cx^2$ is differentiable at $x=0$.

Question

Let $r>0$ and $f: K_r(0)\to \mathbb R$ with

$$\exists C>0\;\forall x\in K_r(0): |f(x)|\le Cx^2 $$

Show that $f$ is differentiable at $x=0$.

Answer 1

By the given property $|f(0)|\le C0^2=0$ which implies that $f(0)=0$. Hence, for $h<r$ $$\frac{f(0+h)-f(0)}{h}=\frac{f(h)}{h}\le \frac{|f(h)|}{h}\le \frac{Ch^2}{h}=Ch$$ Take the limit as $h$ goes to $0$ on both sides to conclude that $$\lim_{h\to 0}\frac{f(0+h)-f(0)}{h}=0$$ Hence $f'(0)$ exists and is equal to $0$.