Past exam of the mathematics course of the University of Cambridge / 2014 / ib / Paper 2 / 4E / i / Solution

False. The constant continuous function $f:\mathbb{R}\to \mathbb{R}$, $f(x)=0$, sends the open set $\mathbb{R}$ to $\{0\}$, which is not open. Therefore continuity alone does not imply an open map.