Preserve adherent, boundary point [closed]

Problem: Let $f:X\to Y$ be continuous and bijective, for all $S\subset X$. Prove or disprove (a) $f$ preserve adherent point, ie $f(\overline{S})\subset \overline{f(S)}$; (b) $f$ preserve boundary point, ie $f(\partial S)\subset\partial (f(S))$.

Here is my attempt:

(a). True, $\forall x\in\bar{S},\forall r>0,B(x,r)\cap S\neq\emptyset\implies f(B(x,r))\cap f(S)\neq\emptyset\implies B(y,r')\cap f(S)\neq\emptyset$ where $y=f(x)$. Hence, we get $\forall y\in f(\bar{S}),y\in\overline{f(S)}$, so proven.

(b). True, $\forall y\in f(\partial S),\exists x\in\partial S$ such that $f(x)=y\implies x\in\bar{S}\cap\overline{X\backslash S}\implies f(x)\in f(\bar{S})\cap f(\overline{X\backslash S})\implies y\in\overline{f(S)}\cap\overline{f(X\backslash S)}\implies y\in\partial(f(S))$, so proven.

I am not sure if my answers are rigorous.

Thanks in advance!