weak derivative

**Veve** · November 20th, 2011, 05:31 PM

I have the function f(x)= x+1 for x greater or equal to 0 and -x-1 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.

**Jose27** · November 20th, 2011, 07:11 PM

Notice that $f$ has classical derivatives on $(-\infty,0)$ and on $(0,\infty)$ , so any weak derivative of $f$ must coincide, on these intervals, with the strong derivative, so the only possible choice of derivative is $g(x)=1$ if $x>0$ and $g(x)=-1$ for $x<0$ . Now take $h \in C_c^{\infty}(\mathbb{R})$ , we calculate

$\int_{\mathbb{R}} fh'= 2\left( xh(x)|_{0}^{\infty}-\int_0^{\infty} h(x)dx\right) - 2h(0)$
$\int_{\mathbb{R}} gh= 2\int_0^{\infty} h(x)dx$

By the definition of weak derivative, the sum of these two quantities must be zero, so we arrive at $h(0)=0$ . Since $h$ was arbitrary we get our contradiction. Notice that the contradiction works only because we allow the support of $h$ to be outside the sets where we know the function is differentiable.

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