TITLE: It there an example of sum and product of continuous functions is not continuous?
QUESTION [7 upvotes]: The sum and product of two continuous functions is continuous

I can prove this easily when the space is metrizable, but I don't get it when the space is non-metrizable. Is there a counterexample of this? or it is true for all topological spaces those have a binary operation?
edit: It it true even if the space has a non-contionuous binary operation?

REPLY [12 votes]: If $X$ is any space and $f,g: X \to \mathbb R$ are any two continuous functions, then $(f,g): X \to \mathbb R \times \mathbb R$ is continuous by definition (or construction) of the product, and $+: \mathbb R^2 \to \mathbb R$ is a continuous function. Then $f+g = {+} \circ (f,g)$, and a composition of continuous functions is continuous.
Similarly for $\times$, of course.
Edit: or replace $\mathbb R$ with $\mathbb C,\mathbb Q$, or any topological ring -- that is, a ring so that $+,-,\times$ are continuous.