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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?