Continuity and adherence in topology

Question

Do we have

$f:(E,\tau)\to(F,\sigma)$ continuous if and only if $\forall A\subset E, f(\overline{A})= \overline{f(A)}$

or just: $f:(E,\tau)\to(F,\sigma)$ continuous if and only if $$\forall A\subset E, f(\overline{A})\subset \overline{f(A)}$$

Answer 1

The later. For instance, take $f\colon\mathbb{R}\longrightarrow\mathbb R$ defined by $f(x)=\frac1{1+x^2}$ and $A=\mathbb R$. Then$$f\left(\overline A\right)=f(\mathbb{R})=(0,1],$$whereas $\overline{f(A)}=[0,1]$.