>>16770547
Hmm, this is a good point.

For definitions like this I imagine it more like we have a list of properties we want this object to have based on intuitive conception, like how we want manifolds to look locally the 'same' as some section of R^n, topologically the idea of which is a homeomorphism. Its how we can guarantee transition maps are at least 'continuous'. Asking the real 'importance' of a property is good but sometimes, not all properties are spelled out in their use (see Proposition 4 below)

It might not be as 'spelled out' here: but differentiability is only defined on open sets. So, as W is open, and our function is defined on y^-1(W), we can only talk about differentiability of x^-1y if y^-1(W) is open. This is where x and y inverse being continuous is needed.

In any case, for the concepts used, Proposition 4 kind of highlights how x and y inverse being continuous is a 'superfluous' condition which is another reason I don't think it was explicitly used. Not saying this doesn't make your criticism valid but the proof at least is correct.

There's some deeper ideas going on here: first, abstract manifolds simply utilize charts as homeomorphisms. It's all topological. They are homeomorphisms so to as to guarantee the transition map is a homeomorphism. As this defined on an abstract topological space M, we have no 'R^n' to talk about differentiability with x and y alone. Next, to define a smooth structure, require x^-1y to be smooth for all parametrizations x and y. So, this is why the definition is given as so based on x and y being homeomorphisms even if it falls out from the definition he uses in this specific instance. (think of his definition as the pragmatic case for surfaces)