Only the closure containment survives for a general continuous map; interior and boundary containments can fail

Let X and Y be topological spaces and $f:X\to Y$ be continuous. (1) $f(\bar A)\subset\overline{f(A)}$, (2) $f(\mathring A)\subset \mathring{f(A)}$ and (3) $f(\partial A)\subset\partial f(A)$. For every subset A of X: