Premise: I am not a mathematician, nor an expert in topology. So sorry for the dumb question, if dumb it is.
Consider a smooth function $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$, and a closed subset $A \subset \mathbb{R}^n$.
We have that $F(\delta A) \in \delta(F(A))$ (see e.g. A homeomorphism maps boundary in boundary?).
I am very curious about the other inclusion $\delta(F(A)) \in F(\delta A)$. Intuitively, it looks clear to me that if $F$ is regular enough, then also this should be true - of course for the $R^n$ to $\mathbb{R}^n$ case only. However, I can not figure out a proof, nor a counterexample.
Do you have any suggestions?
The notation should be $F(\partial A)\subseteq \partial(F(A))$, since these are two sets one contained in another. Also, smoothness does not really play a role here, since the boundary (defined as the closure minus the interior) is a topological notion.
The other inclusion is not true in general. For example, let $f\colon \mathbb{R}\to \mathbb{R}$ given by $f(x)=0$ for all $x\in \mathbb{R}$. Then $\partial \mathbb{R}=\varnothing$, whereas $\partial \{0\}=\{0\}$.
This function is constant, so it is also smooth, in case you are interested in the smooth case. The analogous counterexample works in higher dimensions too.