TITLE: Openness and differentiation
QUESTION [5 upvotes]: Given that $A$ is an open set in $\mathbb R^n$ and $f:A \to \mathbb R^n$ is differentiable, and its derivative is non-singular at every point in $A$, prove that $f(A)$ is open in $\mathbb R^n$
Note $f$ is differentiable, not continuously differentiable.

REPLY [2 votes]: The answer is given in this excellent post by Terence Tao 
https://terrytao.wordpress.com/2011/09/12/the-inverse-function-theorem-for-everywhere-differentiable-maps/