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Consider the metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ and a function $f:X \rightarrow Y$ for every $A \subseteq X$.

I was able to prove that $f$ is continuous iff $f(\overline{A}) \subseteq \overline{(f(A)}$. Now I am looking for any examples for which $f(\overline{A}) \subsetneq \overline{f(A)}$ but I cannot find any.

Also, is it true that on the otherhand the statement "if $f$ is continuous then $(f(A))^{\mathrm{o}} \subseteq f(A^{\mathrm{o}})$" does not hold either?

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Take $\exp\colon\mathbb{R}\longrightarrow\mathbb R$ and take $A=\mathbb R$. Then$$f\left(\overline A\right)=f(\mathbb{R})=(0,+\infty)\neq\overline{f(A)}.$$

On the other hand, if $f\colon\mathbb{R}\longrightarrow S^1$ is the function defined by $f(t)=\bigl(\cos(t),\sin(t)\bigr)$ and $A=(0,2\pi]$, then $(1,0)$ belongs to the interior of $f(A)$, but not to the image of the interior of $A$.

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  • $\begingroup$ Thank you for your insights, these are really clear examples. $\endgroup$ – simp Jun 17 '17 at 20:33

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