 Boundedness extends across closure under a continuous map

For a continuous map into a metric space, $f(\overline{A})\subseteq \overline{f(A)}$: an open neighborhood of an image point pulls back to a neighborhood meeting $A$. A bounded set lies in a closed ball, and its closure lies in the same ball because distance from the centre is a continuous function. The displayed boundedness conclusion follows. Neither compactness of the domain nor uniform continuity of the map is needed.