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.

In plain English:


*

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


Not necessarily.  We know that $f$ is strictly increasing, but that does not imply that it is continuous.
Counter-example: Define $f$ as any strictly increasing, non-continuous function.
In other words, stating that the input to $f$ "changes smoothly" (i.e. $g$ is continuous) states nothing whatever about the output of $f$.



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


Not necessarily.  $g$ could be any arbitrary function; it may not be continuous.
Describing a function as "continuous" states that if the input changes smoothly, the output changes smoothly.  If the input (in other words, the output of $g$) jumps around arbitrarily (discontinuous), the output of $f$ may not change smoothly.



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


Yes, but note that the information provided at the beginning of the question, that $f$ is strictly increasing, is necessary to prove this point.
If this restriction is omitted, the following would be a counter-example:
$g(x) = \begin{cases} x \le 0.5 && 0.2 \\ x \gt 0.5 && 0.8 \end{cases}$
$f(x) = 4x^2 - 4x +1$
Note the following attributes of the above function definitions:

*

*$f$ is continuous.


*$f \circ g$ is continous.


*$g$ is not continuous.
