How do you show that $F$ as defined above is continuous? I do understand intuitively why $F$ is defined the way it is and I can understand intuitively why $F$ is continuous. But I'm not sure how you would actually prove that this function is continuous.
The best I can do is prove that if you fix $s \in [0,1]$, then the restriction of $F$ to the set $I \times\{s\}$ is continuous. I thought about then using the Glue lemma to glue all of these "horizontal strips" together, but there are infinitely many and they are all closed but not open subsets of $I\times I$, and the glue lemma only works for finite unions of closed sets.
I've looked at other sources, and they all give this function $F$ (or something similar at least). But no source actually proves that $F$ is continuous. Am I missing something obvious as to how you can easily show that this function is continuous?
EDIT: Sorry, just to add, the source of the above text is "A basic course in topology" by William S. Massey