Construction of a no constant but vanishing derivative function

Question

This is a problem from Rudin RCA 7.7:
Construct a continuous monotonic function $f$ on $\Bbb R^{1}\equiv\Bbb R$ so that $f$ is not constant on any segment although $f^{'}(x)=0$ $m$-a.e.

Some of my thoughts:
I consider the function $F(x)$=$m([0,x] \cap$ A) for measurable set A,then we have:

$F^{'}(x)=1$ a.e. on $A$
$F^{'}(x)=0$ a.e. on $A^{c}$.

However, if $m(A)=0$, then $F$ is constant on any segment; if $m(A)\neq 0$, then $F^{'}(x)$ can not be zero a.e. on $\Bbb R$. This makes me confused, thanks for your help!

There is a similar problem here, however how to variant it to the whole $R$? Constructing a strictly increasing function with zero derivatives

Does this answer your question? Constructing a strictly increasing function with zero derivatives — r9m, Feb 28 '21 at 04:36
@r9m emmm...I think this solution may not be the answer Ruin wanted cause Cantor-Vitali function has not be taken in his book.... — ymm, Feb 28 '21 at 04:39
I don't know any examples which are intrinsically different from the Cantor function .. only linked for reference. :) — r9m, Feb 28 '21 at 04:50
@r9m, this example does not satisfie cause this is not defined on R... — ymm, Feb 28 '21 at 07:18
this example does not satisfie cause this is not defined on R --- Have you ever heard the phrase "extend by periodicity" or a similar such phrase? This is a fairly well known notion in analysis. Not that anyone would expect you to have seen this, but note that Rudin himself uses (essentially) this phrase in this paper (p. 321, line -3): "... and define $f(x)$ for all other $x$ by periodicity". — Dave L. Renfro, Mar 01 '21 at 09:37

Construction of a no constant but vanishing derivative function

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