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 $g(x)=0{}{}{}{}{}{}$.

  • $\begingroup$ great Sir...I am always missing such examples... $\endgroup$ – MathLover Dec 2 '18 at 9:29
  • $\begingroup$ But Sir If We want both map as surjective map then is it possible to such construct map? $\endgroup$ – MathLover Dec 2 '18 at 9:31
  • 2
    $\begingroup$ @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. $\endgroup$ – Arthur Dec 2 '18 at 9:36

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