I'm facing this exercise; I'm new to distribution theory so I have lots of difficulties: let $f(x):R \rightarrow R, f(x) =4|x-5|$ for every $x \in R$. Are these statements TRUE or FALSE ?
1) $T'_f=4 \delta_5$ ?
2) $T''_f=8 \delta_5$ ?
3) $T''_f=5 \delta_4$ ?
4) $T'''_f= T_0$ ?
From theory, in this case $(T_f)'=T_{f'}$ because $f$ is absolute continuous. And so I can chose a $\phi \in D'(R)$ and have $<T'_f,\phi> = -<T_f,\phi'>=- \int f \phi'$
So, $f' = 4$, and simply here I'm stuck, I don't know how to apply, seems the derivative of $f$ is vanishing everything away (and in this way $f''=0$ so it's even worst). What am I missing?
It s not true that $f'=4$.
Answer for 1): the equation $T_f'=4\delta_5$ means that $-\int 4|x-5|\phi(x)dx=4\phi(5)$ for every test function $\phi$. This is false. For example you can have a non-negative test function vanishing at $5$ which is not identically $0$. In that case the right side is $0$ but the left side is $<0$.