4

If f is a strictly positive Riemann integrable function defined on a closed and bounded interval. Then, is it necessary that f has strictly positive Riemann integral value? f need not be continuous.
I was trying to construct some example similar to Thomae function but such that it is strictly positive but in that case I was getting integral to be strictly positive.

Martin Sleziak
  • 53,687
akansha
  • 1,877
  • 1
    Just intuitively, if for all $x$ the values $f(x)$ lie above the $x$-axis, then their average value (this is what the integral gives you) cannot be below the $x$-axis – Keeran Brabazon Sep 20 '13 at 16:20
  • See also: http://math.stackexchange.com/questions/996282/is-there-a-function-f-gt-0-such-that-int-f-dx-0 – Martin Sleziak Oct 29 '14 at 07:30

1 Answers1

5

Hint: The set of discontinuity points of a Riemann integrable function has Lebesgue measure zero. In particular, it is continuous at some point of the interval.

njguliyev
  • 14,473
  • 1
  • 26
  • 43