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.

I think, for this kind of question, it's best to go with your gut and see if you can't produce a few examples that corroborate or refute it before trying to prove it. Can you think of any examples or counterexamples to either (1) or (2)? Your proposed answer to (3) is indeed correct, so let's focus on (1) and (2).
For (2), the only things we're given are that $f$ is continuous and strictly increasing. But $g: [0,1] \to [0,1]$ could be pretty much anything! So, your first hunch should probably be "no, $f \circ g$ need not be continuous." Can you find a counterexample?
For (1), try to use the same type of reasoning.
