Could you give me a hint on this problem?
Show that $f:A\subset\mathbb{R}^n\longrightarrow \mathbb{R}^m$ is continuous if and only if for every subset $B$ of $A$ , $f(A\cap\overline{B}) \subset \overline{f(B)}$.
Currently I know these definitions of continuity:
$(\rm i)$ In terms of pre-image of open sets.
$(\rm ii)$ In terms of $\epsilon$-$\delta$
$(\rm iii)$ In terms of convergent sequences.
By the statement of the problem I could guess the definition to use in this case is the first one. Maybe you could tell me how to start so I can give it a try, thanks in advance.