Let K be a D-domain and let f:K--R be differentiable on the interval [a,b] as a subset of K (a< b).

f(x) = { 1 if x>0

{ -1 if x<0

is the derivate of f(x) = |x|.

Why doesn't this contradict Darboux's theorem?

I don't understand how this DOESN'T contradict Darboux. There is discontinuity in the f'(x) graph because it jumps from -1 to 1. Also it violates the IVT because there is no u for which f(a)<u<f(b).

Please help