# Does joint continuity imply separate continuity?

I know that separate continuity does not, in general, imply joint continuity but does the converse hold? Given $$X,Y,Z$$ topological spaces, if $$f: X\times Y \to Z$$ is continuous ($$X\times Y$$ with the product topology) does it follow that $$f$$ is separately continuous?

• Is this your question : "Is $f_{X,y_0}:{X\times \{ y_0\} }\to Z$ continuous ? " Apr 9, 2020 at 15:13
• @JoJomax hey, i think you mistyped it because it is not in mathematical format but i think it is! Apr 9, 2020 at 15:15
• Yes . $f_{|_{X\times \{ y_0\} }}$ is continuous . Apr 9, 2020 at 15:17

Yes, if $$f : X \times Y \to Z$$ is continuous, then for each $$(x_0, y_0) \in X \times Y$$, $$f(x_0, y)= f\restriction_{\{x_0\} \times Y}$$ is continuous as a map from $$Y \to Z$$ and so is $$f(x, y_0) = f\restriction_{ X \times \{y_0\}}$$ as a map $$X \to Z$$. The maps $$x \to (x,y_0)$$ and $$y \to (x_0, y)$$ are always embeddings.