**NOTE**. This is not really the question I wanted to ask. Somehow I forgot to mention that I am assuming $f$ is continuous. However, since Iosif's answer has been well-received I have left this question as it was and opened a new one with the right hypotheses. You can find it here A seemingly trivial property of continuous functions differentiable at the origin (PART 2).

Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a function such that $F(0)=0$, $F$ is differentiable at $0$ and $DF(0)$ is invertible. Is it true that for all $\epsilon>0$ there is $\delta>0$ such that $$ DF(0)(B_{(1-\epsilon)\delta})\subset F(B_\delta)\subset DF(0)(B_{(1+\epsilon)\delta}), $$ where $B_r$ is the ball of radius $r$ centered in $0$?