# Basic Question about absolutely continuous functions

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## Main Question or Discussion Point

My question is maybe elementary but I don't know the answer. I have a function f absolutely continuous in (a,c) and in (c,b), f continuous in c. Is f absolutely continuous in (a,b)?
I think the answer is negative but I can't find a counterexample. I really apreciatte your help.

mathman
Possible counterexample. two semicircles (diameters on top), touch at c. The derivative is plus infinite to the left and minus infinite to the right (of c).

mathwonk
Homework Helper
i think this looks true. given epsilon you have to find a delta so that on any finite sequence of non overlapping intervals of total length less than delta, the total variation of the function is less than epsilon. try considering two cases, how small does delta have to be if none of the intervals contain c and how small does it have to be if some interval does contain c, then try to satisfy both conditions at once.

Svein
An absolutely continuous function can be expressed as the difference between two non-decreasing continuous functions (and vice versa). Apply that.

S.G. Janssens
An absolutely continuous function can be expressed as the difference between two non-decreasing continuous functions (and vice versa). Apply that.
This is not true.

Svein
This is not true.
G. Ye. Shilov: Mathematical Analysis (A Special Course). Pergamon Press 1965, page 306.

S.G. Janssens
G. Ye. Shilov: Mathematical Analysis (A Special Course). Pergamon Press 1965, page 306.
The "vice versa" is not true, because there exist continuous, non-decreasing functions that are not absolutely continuous.

Let $f : [0,1] \to \mathbb{R}$ be the Cantor function. Then $f$ is continuous, non-decreasing, but not absolutely continuous. The zero function is continuous and non-decreasing. The difference $f - 0$ is not absolutely continuous.

FactChecker
Gold Member
That dang Cantor. If it wasn't for him, math would be much simpler. ;>)

mathwonk
Homework Helper
i believe Svein is thinking of (continuous) functions of bounded variation, a larger class than absolutely continuous ones.

Last edited:
Delta2
Homework Helper
Gold Member
The "vice versa" is not true, because there exist continuous, non-decreasing functions that are not absolutely continuous.

Let $f : [0,1] \to \mathbb{R}$ be the Cantor function. Then $f$ is continuous, non-decreasing, but not absolutely continuous. The zero function is continuous and non-decreasing. The difference $f - 0$ is not absolutely continuous.
I don't have the book Svein suggests to check it myself but perhaps the theorem/lemma is about non-decreasing AND non-identically zero functions.

S.G. Janssens
Just replace the $f$ from before by the sum $g + h$, where $g$ is the Cantor function and $h$ is any absolutely continuous, non-decreasing function that is not identically zero. Then $g + h$ is continuous, non-decreasing but not absolutely continuous. The difference $(g + h) - h$ is not absolutely continuous.