so if f' is 0 at a |f|' is zero at a too. The other direction is false, i.e. if |f| is differentiable at a, f could fail to be differentiable at a. I'm still not sure what you mean by D though, so I'm not sure if this is what you want...
Well the original problem appears to be wrong. If you mean " iff |f| is differentiable at a" then it is false by the example I gave in post #13. If you mean "if then |f| is differentiable at a (with derivative 0)" then I see post #10.