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  • 1 Post By HallsofIvy

Thread: continuity

  1. #1
    Junior Member
    Jan 2013
    kolkata ( India)


    let f:U->V & g:V->W be two functions such that f and gf is continuous. Is g continuous? Assuming the composition is well defined. Here U,V,W are topological spaces.
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  2. #2
    MHF Contributor
    Sep 2012

    Re: continuity

    Hey swarna.

    It should be continuous if gf is preserved to be continuous. If you are maintaining continuity then it means that any deformation of the space has to be continuous if you start off with something continuous.

    An easier way to think of it is this: pretend that you are starting at a situation where f is already applied. Now you know that applying g will result in a continuous deformation since gf is continuous. Therefore you have to conclude that applying g as if you were starting from something where f was applied is continuous.

    You could do this by a proof of contradiction by assuming that in space V (which is continuous), a deformation to space W which is discontinuous provides a continuous map which should lead to a contradiction.
    Thanks from swarna and topsquark
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  3. #3
    MHF Contributor

    Apr 2005

    Re: continuity

    No, g does NOT have to be continuous.

    Let U= V= W= R. Let f(x)= 0 for all x, g(x)= 0 if x is rational, 1 if x is irrational. The f is clearly continuous for all x while g is discontinuous for all x. What is g(f(x))? Is it continuous or discontinuous?
    Last edited by HallsofIvy; Jun 11th 2013 at 07:10 AM.
    Thanks from emakarov
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