Counter example of continous function such that there is Set S with $f(S)^\circ \subset (f(S^\circ))$

$$S^\circ$$ denotes the interior of a set $$S$$. Is there an example of a continuous function $$f$$ and a set S with $$(f(X))^\circ \not\subset f(S^\circ)$$ ?

I know that$$f(S^\circ)\subset (f(S))^\circ$$ is not always true; for example

$$f(x)=x, ....[0,1]$$

$$f(x)=x-1 ....[2,3]$$

I tried hard but I could not find counterexample for $$(f(S))^\circ\subset f(S^\circ)$$

Any help will be appreciated

This may be overkill, but the cantor function with $$S=$$ the middle-thirds cantor set will work:
2 properties of this function are: $$S^\circ = \emptyset$$ and $$f(S)=[0,1]$$