Example of bounded $f:[0,1]\to\mathbb{R}$ not Riemann integrable but derivative of some function

Question

This problem is taken from Problem 2.4.31 (page 84) from Problems in Mathematical Analysis: Integration by W. J. Kaczor, Wiesława J. Kaczor and Maria T. Nowak.

Give an example of a bounded function $f:[0,1] \to \mathbb{R}$ which is not Riemann Integrable, but is a derivative of some function $g$ on $[0,1]$.

@Chandru: what's lacking? Volterra's function has exactly the properties you request. @Akhil: the link is wrong. — Nate Eldredge, Aug 12 '10 at 22:04

score 8 · Accepted Answer · edited Mar 31 '22 at 08:06

8

I gave an answer to this question on Math Overflow some months ago:

See, in particular, this paper: Goffman, Casper A bounded derivative which is not Riemann integrable. Amer. Math. Monthly 84 (1977), no. 3, 205--206.

edited Mar 31 '22 at 08:06

Martin Sleziak

answered Aug 12 '10 at 22:01

Pete L. Clark

The university link is broken, however. I found a copy on JSTOR instead. – Lei Zhao Mar 13 '21 at 15:12

1 Answers1