
weak derivative
I have the function f(x)= x+1 for x greater or equal to 0 and x1 for x less than 0 ,
I have to prove that f has no weak derivative.
I computed <Df,g>= 2g(0)+\int_0^\infty g(x)dx  \int_{\infty}^0 g(x)dx , where g is a test function. But how do I prove that this is of the form \int_{infty}^\infty h(x)g(x)dx and that Df is not in L^1_{loc}(R)?
Thanks.

Re: weak derivative
Notice that has classical derivatives on and on , so any weak derivative of must coincide, on these intervals, with the strong derivative, so the only possible choice of derivative is if and for . Now take , we calculate
By the definition of weak derivative, the sum of these two quantities must be zero, so we arrive at . Since was arbitrary we get our contradiction. Notice that the contradiction works only because we allow the support of to be outside the sets where we know the function is differentiable.