Are all proper maps continuous Some textbooks define proper maps as continuous maps which have inverse images of compact sets compact. Whereas wikipedia defines them to be having inverse images of compact sets compact. Wikipedia definition does not include continuity? Does continuity follow anyway? I couldn't figure out.
Thanks.
 A: The function $f:[0,1]\to \mathbb R$, 
$$f(x) = \begin{cases} \frac 1x & \text{if }x\neq 0 \\ 0 & x=0.\end{cases}$$
is proper but not continuous. 
A: Any subset of a finite topological space is compact.  So if $X$ is finite and we have a function $f: X \rightarrow Y$, the preimage of any (compact) subset of $Y$ will be compact in $X$.  Thus, to find counterexamples, it suffices to find a finite set $X$ together with a function $f:X \rightarrow Y$ that is not continuous.
Looking at small topological spaces, we can construct such a function explicitly.  For instance, let $X$ be the topological space of $3$ points $\{a, b, c\}$ with open sets $\Big\{\emptyset, \{a\}, \{b\}, \{a,b\}, \{a,b,c\}\Big\}$, and let $Y$ be the topological space of $2$ points $\{x, y\}$ with open sets $\Big\{ \emptyset, \{x\}, \{x, y\} \Big\}$.
Define $f: X \rightarrow Y$   such that:
$$\begin{align}
&a \mapsto y 
\\ &b \mapsto y 
\\ &c \mapsto x 
\end{align}$$
Though $f$ is proper, we can see that $f$ is not continuous since $\{x \}$ is open in $Y$, but $f^{-1}\big(\{x\}\big) = \{c\}$  is not open in $X$.
