1
vote
2answers
68 views

Derivatives of $|x|$

I wanted to calculate the first and second derivatives of the function $f(x)=|x|$, in order to verify that: $$ f'(x)=\frac{|x|}{x} $$ and $$ f''(x)=2\delta(x). $$ Can you help me?
2
votes
4answers
124 views

How can you show that $\delta′=f(0)\delta′−f′(0)\delta$ for a function f that is infinitely differentiable?

Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional. I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation ...
2
votes
0answers
61 views

Distributions - please check my solution

I have to find a derivative in a distributional sense of the following function (known as Cantor's singular function) $$f(x)=\left\{ \begin{array}{l l l l l} 0, & \quad\text{$ x\leq 0 $}\\ 1, ...
1
vote
1answer
64 views

Identify the distrionbutional derivative with classical derivative?

I am reading Rudin's Functional Analysis and got quite confused by his proof for theorem 7.25, which he calls Sobolev's Lemma. In proving the theorem, he defines the function $F$, and calculates its ...
3
votes
4answers
3k views

Which of these two ways to take the derivative of a delta function times another function is correct?

A well known identity of the Dirac delta function is that for any function $f(x)$: $$ \delta(x) f(x) = \delta(x) f(0). $$ If we take the derivative of the right hand side we get: $$ ...