# Counterexample for $f$ is strictly increasing ,$g$ and $f\circ g$ is continous but f is not continous

I wanted to find counterexample

Counterexample for $$f$$ is strictly increasing,$$g$$ and $$f\circ g$$ is continuous but $$f$$ is not continuous

Where f and g are function form $$[0,1]\to [0,1]$$

How to approach to find such example

Any Help will be appreciated

Hint: Let $$g(x)=0{}{}{}{}{}{}$$.
• @MathLover No, if $g$ is surjective and continuous, and $f$ is not continuous, then for some $x_0$, we will have $g(x_0)$ be some point of discontinuity of $f$, which means that $f(g(x))$ must be discontinuous at that point. I've glossed over a few details here, but that's the general argument. – Arthur Dec 2 '18 at 9:36