A classical example of a function of two variables continuous in each but which fails to be continuous is this one. I wonder if someone can suggest an example of an associative counterexample of a binary operation continuous in each variable which fails to be continuous (it doesn't have to be on $\mathbb{R}$ but I expect that to be the easiest setting for analytic examples).
Context: I am currently working with topological monoids, and have found (for a fixed multiplication) a canonical topology related to an action of the monoid which guarantees that the multiplication is continuous in each variable. I would of course like the topology to make it fully continuous, but a priori it is not clear that this holds.
A very natural example is addition with $\infty$. Let $X=\mathbb{R}\cup\{\infty\}$, topologized as the one-point compactification of $\mathbb{R}$ (more generally, we could replace $\mathbb{R}$ with any locally compact group that is not compact). Consider the addition operation $+:X\times X\to X$, where any sum involving $\infty$ is defined to be $\infty$. Then it is easy to see that $+$ is continuous separately in each variable, and is associative. But it is not jointly continuous; for instance as $x\to\infty$, $(x,-x)\to(\infty,\infty)$, but $x+(-x)=0$ does not converge to $\infty+\infty=\infty$.