Continuity of composition of continuous functions

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?

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?

Mr Davis 97
Why is ##\log|x|## not continuous?
Because of the vertical asymptote at 0?

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
zwierz
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.