Equivalent definitions of continuity at a point I'm going with a definition of a map over defined on topological spaces $f:X\rightarrow Y$ is continuous at a point $x\in X$ 
is as follows:

$f$ is continuous at each element $x\in X$ if and only if for every 
  $V$ is open in $Y$  with $f(x)\in V$ there is an open set $U_x\subset X$ with $x\in U_x$ 
  and $f(U_x)\subset V$ or equivalently $U_x\subset f^{-1}(V)$.

Let $\overline{A}$ be the closure of a subset $A$
in $X$.

Prove that a map $f$ from a topological space $X$
  to a topological space $Y$ is continuous 
  at a point $x_{0}$ in $X$
  if and only if
  $f(x_{0}) \in \overline{f(M)}$ for any subset $M$ in $X$
  such that $x_{0} \in \overline{M}$.

I'm having trouble how to get started with the problem.
Any help would be much appreciated.
 A: Since you want to show that both definitions are equivalent you have to show that each of them implies the other one.
Let us first assume that $f$ is continuous at $x_0 \in X$ and let $M \subseteq X$ be a subset with $x_0 \in \overline{M}$. Recall that this means that every open subset containing $x_0$ intersects $M$ nontrivially. We want to show that $f(x_0) \in \overline{f(M)}$, so we have to show that every open subset of $Y$ which contains $f(x_0)$ intersects $f(M)$ nontrivially. In fact if $V$ is such an open subset then continuity implies $f(U) \subseteq V$ for some open subset $U$ of $X$ which contains $x_0$. Now use $x_0 \in \overline{M}$ and $f(U) \subseteq V$ to conclude that $V \cap f(M)$ is not empty.
For the other direction, we want to show that if $f$ is not continuous in $x_0$, then we find a subset $M$ of $X$ such that $x_0 \in \overline{M}$ but  $f(x_0) \notin \overline{f(M)}$. That $f$ is not continuous at $x_0$ means that there exists an open subset $V$ of $Y$ such that $f^{-1}(V)$ does not contain any open subset of $X$ which contains $x_0$. You might want to consider $M = X \setminus{f^{-1}(V)}$ now.
A: To prove that continuity implies that $x_0 \in \overline{M} \implies f(x_0) \in \overline{f(M)}$, you should use the definition of continuity: what can you take from it? Does it state something about relationships among sets of $Y$ and $X$? Can you take them and use the closures?
