Originally Posted by

**shmounal** Let X and Y be topological spaces. Show that $\displaystyle f : X \rightarrow Y$ is continuous if and only if

$\displaystyle \forall A \subseteq X, f(\overline{A}) \subseteq \overline{f(A)}$ . By considering the map $\displaystyle f : \mathbb{R} \rightarrow \mathbb{R} , f(x) = x/(1 + x^2)$, show that we

do not expect $\displaystyle f(\overline{A}) = \overline{f(A)}$ .

For the first part I know, f is continuous if and only if the image of the closure of every subset $\displaystyle \forall A \subseteq X$ is contained in the closure of the image which implies the result but don't think this is really me showing this, its just me stating a rule...

For second part I'd have thought we would expect $\displaystyle f(\overline{A}) = \overline{f(A)}$ so *I'm obviously not understanding something properly.*