# How does one show that the fat cantor characteristic function is nonriemann int'able?

 P: 165 I have been thinking about this for quite some time now. When I look at the function that descibes the fat cantor set namely: f(x) = 1 for x$$\in$$F and f(x) = 0 otherwise, where F is the fat cantor set. I wonder, how do I prove that this is non-riemann integrable? I have considered looking at the Riemann-Lebesgue theorem which gets me nowhere. So f is obviously bounded. But isn't this f discontinuous at all x$$\in$$[0,1]? This would imply that the discontinuity points of f need to be a zero set in order for it to be riemann integrable. But isn't the fat cantor set F not a zero set? Any advice?