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Lets say I have a 1-1 strictly increasing point-wise continuous function f: R -> R, and I want to show that the inverse function g: f(R) -> R is also point-wise continuous.

Now from Rudin's PMA Theorem 4.17 it says that if I have a continuous 1-1 mapping of a compact metric space X to a metric space Y. Then the inverse mapping g defined on Y by g(f(x)) = x is a continuous mapping of Y onto X.

Ok so basically I have everything I need accept for one thing, that R is not compact. Ok so first my reasoning goes that given any f(x) ∈ f(R) I simply restrict the domain of f to the mapping f: [x-1 , x+1] -> R and now since f is strictly increasing and by theorem 4.17, since f(x) ∈ f([x-1 , x+1]), I can conclude that g: f(R) -> R is continuous at f(x). And since I can do this for any point in the range of f, I can conclude that g is continuous.

I suppose my main concern is that while I can certainly conclude that g: f([x-1 , x+1]) -> [x-1 , x+1] is continuous and thus continuous at f(x), I'm not entirely convinced that just because I can draw a compact set around any point in R that still doesn't allow me to generalize to saying that g is continuous on all of f(R) although I am inclined to say it is ok since I'm only looking for point-wise continuity. Hope this made since, thanks