Let $X$ be a topological space and $Y$ a uniform space. I would like to understand the proof that, if one endows $Y^X$ with the compact-open topology, if $f_n\to f\in Y^X$ under this topology then $f_n\rightrightarrows f$ uniformly on every compact subset of $X$.
It has been answered on this site and answered on the site Wikibooks, but I have the same issue with both proofs. In the MSE answer, the author writes:
Instead, note that for every $x\in K$ there is an open neighbourhood $U_x$ of $x$ such that $f(U_x)\subseteq B_{\epsilon/3}(f(x))$.
And Wikibooks' proof says:
For each such entourage $[W_x]$ let $U_x$ be an open neighbourhood of $x$ such that $f(U_x)\subseteq W_x(f(x))$
This assumes continuity of $f$ (which admittedly is listed in the given assumptions). However, in Royden's text, it is left as an exercise to show uniform convergence iff. convergence in compact-open topology (where $Y$ is a metric space) with no mention of continuity. The above claims do not in general hold for discontinuous $f$, so I ask: is Royden's claim wrong, or is there a more general proof that does not rely on the continuity property?