1. Let f be a function defined on some neighborhood of x=a. Prove f '(a)=0 iff a is in D(abs. value f)
show me a nice proof..
thanks in advance
$\displaystyle \Big|{|f(a+h)|-|f(a)|\over h}\Big| \le \Big|{f(a+h)-f(a)\over h}\Big|$ 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 "$\displaystyle f'(a)=0$ iff |f| is differentiable at a" then it is false by the example I gave in post #13. If you mean "if $\displaystyle f'(a)=0$ then |f| is differentiable at a (with derivative 0)" then I see post #10.