Is there a way to define a dirac delta of the following form:
$$\delta(x_0-f(x))$$
where the function $f$ is non-invertible and has poles in its derivatives.
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Maybe for some values of $x_0$ it could make sense but I don't think so in general. – PC1 Nov 28 '22 at 00:04
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$\delta(1-\log |x|) = \lim_{n\to \infty} 2n 1_{|1-\log |x| |<1/n}$ doesn't pose much problems – reuns Nov 28 '22 at 00:25