Prove that the function f defined on [0, 1] by f(x) = sgn(sin(1/x)), 0<x<=1 and f(x)= 0, x = 0 is Riemann integrable on [0, 1], where sgn denotes the signum function.
I consider the partition {0,1/nπ,1/(n-1)π,...,1/3π,1/2π,1/π,1} and proved that the given function is Riemann integrable on [1/nπ,1] but not able to prove it for [0,1]. Since f is continuous on [1/nπ,1] except at the set of points 1/nπ, 1/(n-1)π, ..., 1/3π, 1/2π, 1/π which have only one limit point, 0, and hence f is Riemann integrable on [1/nπ,1]. But f achieves both 1 and -1 on [0,1/nπ) and is not continuous.
This function is Riemann-integrable by the Lebesgue-Vitali theorem