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This post is discussing the connection between this equation

\begin{equation} \hat{p}(x) = \frac{1}{m} \sum_{i=1}^m \delta(x - x^{(i)}) \tag{3.28} \end{equation}

and this equation

$\widehat {F}_{n}(t)={\frac {{\mbox{number of elements in the sample}}\leq t}{n}}={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} _{X_{i}\leq t}$

This post says

the derivative of a step function is the Dirac delta.

Consider this step function,

\begin{equation} \hspace{50pt} u(x) = \left\{ \begin{array}{l l} 1 & \quad x \geq 0 \\ 0 & \quad \text{otherwise} \end{array} \right. \hspace{50pt} \end{equation}

Per this post, the function above is not differentiable at $=0$.

Function is constant in (0,∞) and (−∞,0).The derivative is therefore 0.

Is it reasonable to claim that the derivative of a step function is the Dirac delta?

JJJohn
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    Read some distribution theory to see why the claim is true in the distribution sense – Conrad Sep 20 '19 at 01:29
  • Yes agree with what Conrad said. Read about weak derivatives. The usual derivative here does not exist, however you can define a weak derivative. – fGDu94 Sep 20 '19 at 02:29
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    It depends on whether you are a mathematician or a physicist. Physicists pay their functions more, and functions work harder for them (apologies to Lewis Carroll). For physicists, all functions are uniformly continuous and infinitely differentiable. Any reasonable differentiable approximation to a step function will have a derivative that is a reasonable approximation of a delta function. For many purposes that is good enough. Distributions are the way to formalize this if you are a mathematician. – Ross Millikan Sep 20 '19 at 03:01
  • @Conrad Could this tell me the reason? – JJJohn Sep 20 '19 at 03:26
  • @GeorgeDewhirst Could you please recommend one that you learned? – JJJohn Sep 20 '19 at 12:15
  • Yes you can read about it in Evans PDEs – fGDu94 Sep 20 '19 at 12:43
  • @GeorgeDewhirst "Partial Differential Equations by Lawrence C. Evans"? – JJJohn Sep 21 '19 at 22:26
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    Yep. They will be part of the chapter on sobolev spaces – fGDu94 Sep 22 '19 at 02:53
  • @GeorgeDewhirst Thanks! Is this a complete version of Evans PDEs? There is only one occurence of "Sobolev Spaces" in it, which is "Further Reading" section. Is "H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2012" you actually recommend? – JJJohn Sep 22 '19 at 06:00
  • No no this is an overview, if you send me your email i can point you towards a source, so to speak – fGDu94 Sep 22 '19 at 18:13
  • @GeorgeDewhirst Here it is, [email protected], Thanks a lot! – JJJohn Sep 22 '19 at 22:19

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