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Sorry this is a laymen question. I commonly see references to a function of two variables as being 'jointly continuous' especially in proofs using homotopies. I sometimes get confused as to which type of continuity this refers to - product topology versus standard topology.

For example: When one says that $F(s,t)$ is jointly continuous on $[a,b] \times [0,1]$ does one mean that for every $\epsilon$, there is a $\delta$ such that when $\mid t-t_0 \mid <\delta$,
$\mid F(s,t)-F(s,t_0) \mid$ will be less than $\epsilon$ for all $s \in [a,b]$ ? Or does it mean that $\mid F(s,t)-F(s_0,t_0) \mid < \epsilon$ using the euclidean norm?

dualredlaugh
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  • It means that for each $(s_0,t_0)\in[a,b]\times[0,1]$ and each $\epsilon>0$ there is a $\delta>0$ such that if $(s,t)\in[a,b]\times[0,1]$ and $\sqrt{(s-s_0)^2+(t-t_0)^2}<\delta$ then $|F(s,t)-F(s_0,t_0)|<\epsilon$. – John Dawkins Dec 18 '15 at 18:30

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