Sum of continuous functions is continuous I am studying for a test and then came this question: $X$ and $Y$ are topological spaces and $f:X \longrightarrow \mathbb{R} , g: Y \longrightarrow\mathbb{R} $ are continuous. Prove that $h: X\times Y \longrightarrow \mathbb{R} $ where $h(x,y)=f(x)+g(y)$ is continuous. 

I know that exists a similar question on Math StackExchange but I don't understand that proof.
Hint: $h(x,y) &gt; z$ if and only if there are rational numbers $r, s$ such that $f(x) &gt; r$ and $g(y) &gt; s$ and $r + s &gt; z$.
hint

$$|h (x,y)-h (a,b)|\le |f (x)-f (a)|+|g (y)-g (b)|$$