Is a curried continuous function in Euclidean space continuous? Let $A: [a,b]\times[a,b] \to \mathbb{R}$ be a continuous function. Does it follow that, for each $y \in [a,b]$, the function $$A_y: [a,b] \ni x \mapsto A(x,y) \in \mathbb{R}$$ is continuous (assuming Euclidean metric space)? I believe the answer is yes. To establish continuity of $A_y$, fix $\epsilon > 0$ and $x \in [a,b]$. Now find (by continuity of $A$) a $\delta > 0$ so that $d((x,y), (x',y')) < \delta \implies \lvert A(x,y) - A(x',y') \rvert < \epsilon$. Now, $|x-x'| < \delta \implies d((x,y), (x',y)) < \delta \implies \lvert A(x,y) - A(x',y) \rvert < \epsilon$, as desired. Is this proof correct? Does it generalize to arbitrary topological spaces?

Your proof is correct. The result is true for general topological spaces too. If $A:X \times Y \to Z$ is continuous, where $X,Y,Z$ are topological spaces, then the map $X\
i x \mapsto A(x,y)$ is continuous for all $y\in Y$.