Ok, so basically I am trying to decide whether my mathematics is valid or if there is some subtly which I am missing:(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Point-wise continuity on all of R using compact sets

Loading...

Similar Threads for Point wise continuity |
---|

I Video (analytic continuation) seems to mix 4-D & 2-D maps |

I Integrate a function over a closed circle-like contour around an arbitrary point on a torus |

A Continuous mass distribution |

**Physics Forums | Science Articles, Homework Help, Discussion**