let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?
(a). If $g$ is continuous, then $f\circ g$ is continuous.
(b). If $f$ is continuous, then $f\circ g$ is continuous.
(c). If $f$ and $f\circ g$ are continuous, then $g$ is continuous.
(d). If $g$ and $f\circ g$ are continuous, then $f$ is continuous.
I guessed this
$f$ is strictly increasing $\implies$ $f$ is continuous on $[0,1]$ So, If $g$ is continuous then $f\circ g$ is continuous. Is my approach is correct?
If i am right, why the others are wrong? can you give a counter examples for that?
Best Answer
Only statement (c) is true.
Hint: if $f$ is continuous and strictly increasing, it has a continuous (left-)inverse $f^{-1}$. Consider $f^{-1} \circ f \circ g$.
Counterexamples, in no particular order:
An example of a strictly increasing discontinuous function: $$ f(x) = \begin{cases} x/3 & x \in [0,1/2)\\ (x+1)/3 & x \in [1/2,1] \end{cases} $$