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Almost Smooth Functions 11.1 Exercise 4.4

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Andrew Hardy:
I am unsure of how to answer Exercise 4.4 from Chapter 11.1
What does it mean to write $$ ((f(a-0) -(f(a+0))\delta(x-a) $$ If these are ordinary functions, this looks like $$ f(a) - f(a)\delta(x-a) = 0\delta(x-a)= 0  $$

How do I check this and what am I trying to show?

Victor Ivrii:
Ordinary function is a function in the standard sense, albeit not necessarily continuous. F.e. Heaviside function. As $x\ne 0$ we can differentiate it, producing $\overset{\circ}{f}{}'$, but we also can find it's derivative as a distribution, producing $f'(x)$. What is $\overset{\circ}{f}{}'$ here? And what is $f'$ here?

Let $$f(x)=\left\{\begin{aligned}&\cos(x) &&x>0,\\
&0 &&x\le 0\end{aligned}\right.$$
What is $\overset{\circ}{f}{}'$ here? And what is $f'$ here?

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