let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?

Question

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?

(a). If $g$ is continuous, then $f\circ g$ is continuous.

(b). If $f$ is continuous, then $f\circ g$ is continuous.

(c). If $f$ and $f\circ g$ are continuous, then $g$ is continuous.

(d). If $g$ and $f\circ g$ are continuous, then $f$ is continuous.

I guessed this

$f$ is strictly increasing $\implies$ $f$ is continuous on $[0,1]$ So, If $g$ is continuous then $f\circ g$ is continuous. Is my approach is correct? If i am right, why the others are wrong? can you give a counter examples for that?

Answer 1

If $f$ is strictly increasing, we can guarantee that $f$ has, at most, countably infinitely many points of discontinuity. But strict monotonicity does not imply continuity at all. Consider $f(x)=x/2$ for $x<1/2$ and $f(x)=(1+x)/2$ for $x\ge1/2$.

Answer 2

Only statement (c) is true.

Hint: if $f$ is continuous and strictly increasing, it has a continuous (left-)inverse $f^{-1}$. Consider $f^{-1} \circ f \circ g$.

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Counterexamples, in no particular order:

• $f(x) = x$, $g(x)$ is discontinuous
• $f$ is increasing but discontinuous, $g$ is a constant function
• $f$ is increasing but discontinuous, $g(x) = x$

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An example of a strictly increasing discontinuous function: $$ f(x) = \begin{cases} x/3 & x \in [0,1/2)\\ (x+1)/3 & x \in [1/2,1] \end{cases} $$

Answer 3

For (a), consider $g(x) = x$ and $f$ any discontinuous increasing function (for example $f(x) = \frac{1}{2} + \frac{1}{2}x, x > 0$ and $f(0) = 0$).

For (b), consider $f(x) = x$ and $g$ any discontinuous function.

For (d), consider $g(x) = 0$ and $f$ any discontinuous increasing function.