Suppose $f:[0,1]\to [0,1]$ is a continuous nondecreasing function with $f(0)=0$ and $f(1)=1$. Define $g:[0,1]\to [0,1]$ by $g(y)=\min\{x\in[0,1]\mid f(x)\ge y\}$.
Then:
(A) $g$ is nondecreasing
(B) If $g$ is continuous then $f$ is strictly increasing.
I was able to figure out that (A) is true, however I am unable to prove that (B) is false. It seems to me that if $f$ is not constant over any interval, then $g$ should be continuous and vice-versa.
I think I found a function f which gives a contradiction to (B). We use $f: [0,1] \to [0,1] $ with
$f(x) = \begin{cases} 2x, & \text{if $0 \le x \le \frac{1}{2}$ } \\ 1, & \text{if $\frac{1}{2} \le x \le 1$ } \end{cases}$
f is not strictly increasing but still nondecreasing. In contrast to this g is still continuous, as all the minimas are taking place in $[0,\frac{1}{2}]$