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## Homework Statement

The signum function is defined by$$sgn(t)=\left\{\begin{matrix}-1, \ t<0\\0, \ t=0 \\ 1, \ t>0 \end{matrix}\right.$$It has derivative$$\frac{d}{dt} sign(t) = 2 \delta(t)$$Use this result to show that ##j2\pi \nu S(\nu)=2,## and give an argument why ##S(0)=0.## Where ##S(\nu)## denotes the Fourier transform of the signum function

## The Attempt at a Solution

I know that the Fourier transform of the sign function is:$$S(\nu)=\frac{1}{j\pi \nu}.$$

If we substitute this into ##j2\pi \nu S(\nu),## we get ##2## as expected. But the question wants us to show this result

*without*deriving an expression for the Fourier transform of ##sgn(t).## How can we do this?

Furthermore, if we substitute ##0## in ##S(\nu)## we get ##S(0)=1/j\pi (0) = \infty.## So what does the question mean by giving an argument that ##S(0)=0##?