Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 10F / b / Solution

This may be false because continuity only forces the continuous image of a compact space $f(X)$ to be compact, not the whole codomain. For example, let $X=\{0\}$, let $Y=\mathbb{R}$, and set $f(0)=0$. The domain is compact and $f$ is continuous, but $\mathbb{R}$ is not compact.