That function doesn't actually have an integral in any sense that I know of. The Riemann integral doesn't work because, for every interval in every possible partition, the function's upper bound will be 1 and the lower bound will be -1.
My original intuition is that every value in [0, 1] will be cancelled by a value in [-1, 0].
My updated intuition to make it more Lebesgue-compatible would be to sort the (infinite) set of values in the interval. This will give the everywhere continuous, linear function from -1 at the minimum coordinate of the interval to 1 at the maximum coordinate.
Hm... I notice that when applying Lebesgue to Dirichlet, it uses countability, and reals aren't countable. I have no clue whether that's a problem here or not.