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## Homework Statement

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.

## 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!