I assume f satisfies some continuity condition, else it's definitely not true. Hint: show that is dense in [0,2pi].
Dear MHF members,
Question. Suppose that is a continuous periodic function with a period of . If is irrational, then for any show that the sum converges to .
Exercise 19 in Chapter 3 of W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3rd edition, 1976.