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In summary, the theorem states that if two continuous functions are composites, then their inverse functions are also composites. However, the inverse function of a non-composite function may not be composite.

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Mr Davis 97 said:

Why is ##\log|x|## not continuous?

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Because of the vertical asymptote at 0?Math_QED said:Why is ##\log|x|## not continuous?

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Mr Davis 97 said:Because of the vertical asymptote at 0?

But how can you apply the theorem you just stated if ##\log(x)## isn't even defined at ##x = 0##? If you start with false assumptions, surely you will get wrong results.

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In this theorem you must also trace carefully domains of the functions. The exact formulation is as follows. Let ##X,Y,Z## be topological spaces and ##f:X\to Y,\quad g:Y\to Z## be continuous functions. Then the function ##g\circ f:X\to Z## is a continuous function.Mr Davis 97 said:I've learned that composition of continuous functions is continuous

In your case ##f(x)=|x|,\quad g(y)=\log y,\quad X=\mathbb{R}\backslash\{0\},\quad Y=\{y\in\mathbb{R}\mid y>0\},\quad Z=\mathbb{R}##

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Mr Davis 97 said:

But remember that the composition may not even be defined at some points, let alone be continuous. Try specifying the theorem a bit more carefully, as Zwierz , I think, suggested. EDIT: You may want to consider the inverse image of open/closed set is open/closed definition as well.

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