# Continuous real-valued functions defined on non-compact spaces

A continuous real-valued function defined on a compact space is bounded and attains its bounds. Is it so that on a non-compact space it is always possible to define a continuous function that does not have these properties? (In fact, this is the case if the non-compact set lies in a finite-dimensional normed vector space, because in this case the set is either not closed or not bounded.)

• $X$ is pseudocompact if image of any continuous function $f \colon X \to \mathbb R$ is bounded. But nothing is said about attainability of its bounds. So author of question speaks about some subclass of pseudocompact spaces, right? Is there a name for such subclass? – Appliqué Apr 25 '13 at 20:38
• Ah, yes, if $f^\ast$ for example is upper not-atteinable bound of $f$, then $x \mapsto 1/(f^\ast-f(x))$ will be unbounded – Appliqué Apr 25 '13 at 21:01
No, it is not: there are $T_3$ spaces on which every real-valued continuous function is constant. E. K. van Douwen, A regular space on which every continuous real-valued function is constant, Nieuw Arch. Wisk. 20 (1972), 143-145, is a particularly nice example.