Naming Functors

Mathematicians are a creative bunch, especially when it comes to naming things. And category theorists are no exception. So here's a little spin on this xkcd comic. It's inspired by a recent conversation I had on Twitter and, well, every category theory book ever.

A function $f$ is said to be continuous if the preimage of any open set is open. Analogously, we might say that a function is measurable if the preimage of a measurable set is measurable. It's not hard to show that if $\{f_n\}$ is a sequence of measurable functions, then sup$\{f_n\}$, inf$\{f_n\}$, limsup $\{f_n\}$ and liminf $\{f_n\}$ are also measurable functions. But here the analogy between continuity and measurability breaks down. It is not true that if each $f_n$ is a continuous function, then sup$\{f_n\}$, inf$\{f_n\}$, limsup $\{f_n\}$ and liminf $\{f_n\}$ are continuous as well. Below is a counterexample - a sequence of continuous functions with a discontinuous supremum!