When is the exponential law in topology discontinuous? By $Y^X$ I mean the space of continuous functions $X \to Y$ with the compact-open topology (by compact I don't require Hausdorff. Consider the exponential law $Z^{X \times Y} \to (Z^Y)^X$ where the map is well-defined a a set map (this is not hard to see). However I don't think it's always continuous, but I don't have any examples of it being discontinuous. I know that $X$ Hausdorff implies continuous. So any counterexample must have $X$ non-hausdorff. 

There is a paper by P. Booth and J. Tillotson, Pacific  J. Math. 
Vol.  88.  No. 1,  198 (downloadable) which discusses  exponential laws $X ^{Z \times Y} \cong (X^Y)^Z$ for various function space topologies,  and topologies on $X \times Y$. 
