The convexity of convex function's range

Question

Given a convex function $f\colon X \to \mathbb R$ with convex domain $X \subseteq \mathbb R^n$, is the range of $f$ a convex set also?

Answer 1

Let $X = [0,1]$ and let $f:X \to \mathbb R$ such that \begin{equation} f(x) = \begin{cases} 1 \quad \text{if } x = 0 \\ 0 \quad \text{otherwise}. \end{cases} \end{equation} Then $f$ is convex, but the range of $f$ is $\{0,1\}$ which is not convex.