# Continuity of composition of continuous functions

• I
• Mr Davis 97
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.

#### Mr Davis 97

I've learned that composition of continuous functions is continuous. ##\log x## and ##|x|## are continuous functions, but it seems that ##\log |x|## is not continuous. Is this the case?

Mr Davis 97 said:
I've learned that composition of continuous functions is continuous. ##\log x## and ##|x|## are continuous functions, but it seems that ##\log |x|## is not continuous. Is this the case?

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

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

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.

Mr Davis 97
Mr Davis 97 said:
I've learned that composition of continuous functions is continuous
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.