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Continuity of the inverse function

  1. Dec 1, 2011 #1
    I'm having a little trouble with something so I am wondering,

    If f is a continuous 1-1 mapping from an open set (a,b) into ℝ then is its inverse function g continuous at all points of the image of f?

    My argument is that g(y) is in (a,b) for all y in the image of f, and g(y)=x for some x in (a,b).
    So we can find a δ>0 so small such that [x-δ,x+δ] is contained in (a,b).

    Hence f([x-δ,x+δ]) is compact and so g is continuous on that set.

    since we can always find δ's small enough for every x then g is continuous at all points in f((a,b)).

    Is my argument invalid anywhere?
  2. jcsd
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