Re: Continuous piecewise monotone and absolutely continuous functions

In article
<21127994.1218121235628.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Mary Kashkova <kashkova@xxxxxxx> wrote:

A function f:[a,b]->R is called piecewise monotone iff there is a finite
partition
a = x_0 < x_1 < x_2 < ... < x_{n-1} < x_n = b of the segment [a,b]
such that on each segment [x_k, x_{k+1}] (k=0,1,...,n-1) f is either
strictly monotone
(increasing or decreasing) or constant.
For example, f(x)={x+1 if x in [0,1], 2 if x in [1,4], 18-x^2 if x in [4,10]}
is piecewise monotone on [0,10].

1. Let f be a continuous piecewise monotone function. Is f absolutely
continuous?

2. Let g be an absolutely continuous function. Is g continuous piecewise
monotone?

For 2: If g is continuously differentiable on [a, b], then g is AC.
Does there exist such a g with g' changing signs infinitely many times?
.

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