The boundary of the image of a relatively compact domain lies in the image of its boundary

Let \(D\subseteq\mathbb C\) be a domain and let \(B\subseteq D\) be open and bounded with \[ \overline B\subseteq D. \] If \(f\) is holomorphic on \(D\), prove that \[ \partial(f(B))\subseteq f(\partial B). \] Show also that the conclusion can fail when \(B\) is unbounded.