Find an example of a continuous function and open (closed) A such that f(A) is not open(closed) for a continuous function and open A such that f(A) is not open. I think $f:R \rightarrow R$, and $f(A)=0 $ for $A \in (0,1)$ works. since a singe point in $R$ is closed. I have no idea to find a continuous function and closed A such that f(A) is not closed.

Let $f(x) = e^{-x^2}$ and $A= \mathbb{R}$.

Then $f(\mathbb{R}) = (0,1]$ which is neither open nor closed. $A$ is both open and closed.