(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if f: [a,b] -> R is absolutely continuous, and E ∁ [a,b] has measure zero, then f(E) has measure zero.

2. Relevant equations

A function f: [a,b] -> R is absolutely continuous if for every ε > 0 there is an δ > 0 such that for every finite sequence {(xj,xj')} of nonoverlapping intervals in [a,b] with ∑|xj'-xj| < δ, ∑|f(xj')-f(xj)| < ε .

3. The attempt at a solution

I think that there is an alternative definition of absolute continuity using countable intervals instead of finite intervals, and if I knew that the set E was countable I think I could go from there...but I don't know that E is countable; I only know that it has measure zero. So I'm not really sure where to start.

Any help would be much appreciated : )

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Absolutely continuous functions and sets of measure 0.

**Physics Forums | Science Articles, Homework Help, Discussion**