let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true? 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?
 A: 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:


*

*$f(x) = x$, $g(x)$ is discontinuous

*$f$ is increasing but discontinuous, $g$ is a constant function

*$f$ is increasing but discontinuous, $g(x) = x$



An example of a strictly increasing discontinuous function:
$$
f(x) = 
\begin{cases}
x/3 & x \in [0,1/2)\\
(x+1)/3 & x \in [1/2,1]
\end{cases}
$$
A: If $f$ is strictly increasing, we can guarantee that $f$ has, at most, countably infinitely many points of discontinuity. But strict monotonicity does not imply continuity at all. Consider $f(x)=x/2$ for $x<1/2$ and $f(x)=(1+x)/2$ for $x\ge1/2$.
A: For (a), consider $g(x) = x$ and $f$ any discontinuous increasing function (for example $f(x) = \frac{1}{2} + \frac{1}{2}x, x > 0$ and $f(0) = 0$).
For (b), consider $f(x) = x$ and $g$ any discontinuous function.
For (d), consider $g(x) = 0$ and $f$ any discontinuous increasing function.
