Composition of functions is continuous? 
If $f$ any $g$ be two functions defined from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Then

*

*if $g$ is continuous, is $f\circ g$ continuous?


*if $f$ is continuous, is $f\circ g$ continuous?


*if $f$ and $f\circ g$ are continuous, is $g$ continuous?

Here, $f\circ g$ implies composition of $f$ and $g$. I think the answer to the third is yes by using the fact that preimage of an open set under a continuous map is open? Any idea .Thanks.

So the third case is shown to be true properly by learnmore. For the counterexample to case 2), maybe we can take $f=x$ for case 2) and $g=\begin{cases} \frac{x}{2} && x\in[0,0.5)\\ \frac{x+1}{2} && x\in[0.5,1]\end{cases}$. Similar type could be given for case 1)
