1. The problem statement, all variables and given/known data Let f(x) = x sgn(sin(1/x)) if x != 0 f(x) = 0 if x = 0 on the interval I=[-1,1] Now I 'm asked to show that f(x) is not piecewise continuous on I and later, I must show that f is integrable on I. 3. The attempt at a solution I am completely lost here and don't know how to proceed... I think we should first divide the partition in some regular subpartitions so that x=0 is excluded (the only possible discontinuity?): P1=[-1, -e/100] and P2=[e/100, 1], and then try to show that continuity on the interval P3=[-e/100, e/100] is impossible but that we still have uniform continuity on P1 and P2? Should I be looking in that direction? Or should I look at it in a complete different way? Thank you for the help!