Is every function that can be integrated using Riemann integral must be monotonic function?

Question

If false then is there an example.

Do you know any examples of Riemann integrable functions? Are they all monotonic? — Noah Schweber, Apr 08 '16 at 19:14

Patrick Stevens · Accepted Answer · 2016-04-08T19:04:29.807

4

No. $\sin(x)$ is Riemann integrable between $0$ and $\pi$. In fact, any non-monotone continuous function on a compact set.

An example of a nowhere monotone continuous function is given in this answer. (A function is nowhere monotone if there is no nonempty open set on which it is monotone.)

edited Apr 08 '16 at 19:04

answered Apr 08 '16 at 19:02

Patrick Stevens

36,135

Great, Thanks for help. – MrDi Apr 08 '16 at 19:03

Is every function that can be integrated using Riemann integral must be monotonic function?

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