But it seems to restrict to PL manifolds. Can you do it without linearity? And, is it easy to do this as n grows, say for n=17, or so? .How would you do it with, say, an S^n which is not a linear space? EDIT: I know I asked about ##\mathbb R^n ## , but I am curious about the more general case. And, I don't know if I unde understood you correctly, but isn't a product function (fxg)(x,y) defined as f(x)g(y), so that the codomain is the Reals, and not ##\mathbb R^n ##? And if you consider pairs f,g of homeomorphism andas ben said, i would try something like (a,b)-->(a+b,b) on the upper half plane, and (a,b)-->(a,b) on the lower half plane. on the x axis these agree since then b=0. i.e. this seems not differentiable along the x axis. i.e. ask yourself what the best linear approximation should be say in a neighborhood of (0,0)?
or, even easier, just take products of examples from R^1.
defining h: ## \mathbb R \times \mathbb R : h(x,y)=(f(x), g(y))##, is that necessarily a homeomorphism?EDIT 2: Ah, nevermind, we do the same thing for any ##\mathbb R^n,## we just partition into "North and South"