Example of strict inclusion in continuity condition $f(\overline{A})\subseteq \overline{f(A)}$ One definition of continuity is the condition
$$f(\overline{A})\subseteq \overline{f(A)},$$
for all $A\subseteq X$.
To understand this condition better, I tried to find an example of a real-valued function $f\colon\mathbb{R}\to\mathbb{R}$ and a set $A\subseteq\mathbb{R}$ where this inclusion is strict, but I couldn't find one. 
If we let $\mathbb{R}_d$ be the real line with the discrete topology, then the identity map $I\colon\mathbb{R}_d\to\mathbb{R}$ and $A = (0,1)$ is an example with strict inclusion. But this example uses a nonstandard topology on $\mathbb{R}$.


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*My question: Is there an example of a continuous function $f\colon\mathbb{R}\to\mathbb{R}$ and set $A\subseteq\mathbb{R}$ with the usual topology on $\mathbb{R}$ such that 
$$f(\overline{A})\neq \overline{f(A)}?$$


Yes. $\;\;\;$ $\hspace{.04 in}f(x) = \dfrac1{1\hspace{-0.05 in}+\hspace{-0.04 in}\left(\hspace{-0.02 in}x^2\hspace{-0.04 in}\right)} \:$ and $\: A = \mathbb{R}$
