Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Show: $f(\overline{S})\subseteq \overline{f(S)}$ for $S\subseteq \mathbb{R}$ 

(Note: $\overline{S}$ denotes the closure of S; $\partial S$ denotes the set of boundary points of S)

This is my working:

Let $y\in f(\overline{S})$

Then $\exists x\in \overline{S}$ such that $f(x)=y$

$\Rightarrow \exists x \in S$ or $\exists x \in \partial S$ such that $f(x)=y$

$\Rightarrow f(x) \in f(S) $ or $f(x)\in f(\partial S)$

$\Rightarrow f(x) \in f(S) \cup f(\partial S)$ 

$\Rightarrow y\in f(S) \cup \partial f(S) $

$\Rightarrow y\in \overline{f(S)}$

However, I wonder is $f(\partial S)=\partial f(S)$ ?? (If yes, how to prove?) 

Is there any other alternative proof for this question?