Let $f:\mathbb R^n\rightarrow \mathbb R^m$ and suppose there is a constant M such that for $x\in \mathbb R^n,~~||f(x)||\leq M||x||^2$
Prove that f is differentiable at $x_0=0$ and that $Df(x_0)=0$
Please, help

Question

Let $f:\mathbb R^n\rightarrow \mathbb R^m$ and suppose  there is a constant M such that for $x\in \mathbb R^n,~~||f(x)||\leq M||x||^2$
Prove that  f is differentiable at $x_0=0$ and that $Df(x_0)=0$
Please, help

zzr0ck3r · Answer

which norm is this?

loser66 · Answer

vector space, I think!! That's all I have from the problem.

zarkon · Answer

I assume you mean $x_o=(0,0,\cdots,0)$  (an n-tuple)

clearly $f(x_0)=0$ since $$\|f(x_0)\|\leq M\|x_0\|^2=0$$

then just look at 

$$\frac{\|f(x)-f(x_0)\|}{\|x-x_0\|}=\frac{\|f(x)\|}{\|x\|}\leq M\|x\|$$

and let $x\to x_0$

loser66 · Answer

Thank you so much. I think I get it. :)