Uniform continuity and bounded variation does imply absolute continuity? I'm studying absolute continuity. We see that a absolute continuity implies uniform continuity and also see that a function which is uniform continuous but not bounded variation is not absolute continuous.So Uniform continuity and bounded variation does imply absolute continuity?

This is false. The Cantor function $f(x)$ is a counterexample: it is uniformly continuous and of bounded variation, yet it is not absolutely continuous.


*

*Uniformly continuous: $f$ is continuous on the compact set $[0,1]$

*Bounded variation: $f$ is increasing and bounded

*Not absolutely continuous: $f'=0$ almost everywhere, so $f(x)\not=\int_0^x f'(x)\ dx$.
