Continuous function and open set I'm trying to prove: 
If $f:A\subset \mathbb{R}^m \to \mathbb{R}^n$ and $A$ is open, then the following statements are equivalent:
1) $f$ is continuous on $A$.
2) $f^{-1}(V)$ is open in $\mathbb{R}^m$, for all $V\subset \mathbb{R}^n$ open.
Thanks for your help.
 A: This is my answer:
1) $\to$ 2). 
If $f^{-1}(V)=\emptyset$ the answer is trivial.
Let $a\in f^{-1}(V)$. Particularly $a\in A$. As $f$ continuous in $A$ then $\forall \varepsilon>0 \exists\delta>0$ such that, if $||x-a||<\delta \implies ||f(x)-f(a)||<\varepsilon$. i.e:
if $x\in \mathbb{B}_{\delta}(a) \implies f(x)\in \mathbb{B}_{\varepsilon}(f(a))$
$\implies f(\mathbb{B}_{\delta}(a))\subset \mathbb{B}_{\varepsilon}(f(a))$
$\implies \mathbb{B}_{\delta}(a)\subset f^{-1}(\mathbb{B}_{\varepsilon}(f(a)))$
$\therefore f^{-1}(V)$ is open.
2) $\to$ 1).
Let $\varepsilon>0$ and $a\in A$. $V$ is open then exists $\mathbb{B}_{\varepsilon}(f(a)))$ 
open. Now $a\in f^{-1}(\mathbb{B}_{\varepsilon}(f(a))))$ is open. i.e:
$\exists \delta$ such that $\mathbb{B}_{\delta}(a))\subset f^{-1}(\mathbb{B}_{\varepsilon}(f(a))))$ (you can prove that this is true)
$\implies f(\mathbb{B}_{\delta}(a)))\subset \mathbb{B}_{\varepsilon}(f(a)))$
$||x-a||<\delta \implies ||f(x)-f(a)||<\varepsilon $
$\therefore f$ is continuous.
A: Assuming your using the definition of continuity that the inverse image of an open set
 is open, one way of doing the proof is:
i) Show that if the restriction of $f\colon \mathbb R^m \rightarrow \mathbb R^n:$ to any open
   subset of the domain is continuous, then f is globally-continuous, meaning f satisfies
   condition #2. For this, you can (Let X be the domain, mapping into Y to simplify; 
   result generalizes to any two topological spaces X,Y anyway.)
   consider an open subset  W of Y, and consider $f^{-1}(W)$ . By def. , f is continuous
   if $f^{-1}(W)$ is open in the subspace A of X .
2) If we use 2 for the definition of continuity, assume $f:X \rightarrow Y$ is
   continuous, so that  $f^{-1}(W):=V$ is open in X. Can you show V is open in 
   the subspace A?
3) Can you find an outlet, my battery is dy...... 
