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When I plot $$f(x) =\frac{\arctan(\tan(2\pi x))}{2\pi}$$ in WolframAlpha, it gives me an expected sawtooth wave.

When I try the derivative of this function, it gives me $1$. I expect that since there is a discontinuity in this function at $(2k+1)\pi/4$, derivative is not defined in those points in the traditional sense. But I am looking for a derivative expression that can include dirac-delta function. For example, when I try finding the derivative of $\text{Heaviside}(x)$, it gives me a result of $\delta(x)$. I understand that dirac-delta is not a function in strict sense. Can anyone point me towards a solution or generally how to approach this?

Pichi Wuana
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Srini
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  • The Dirac Comb is the distribution you seek as a result. – Mark Viola Aug 03 '16 at 20:05
  • @Dr.MV, I understand there will be a comb at the discontinuity points, but what is the scaling factor for each impulse in that train? Is it 1? How does it depend on the height of the discontinuity? – Srini Aug 03 '16 at 20:20
  • Good question(s). The scaling factor is equal to the size of the jump in the discontinuity. -Mark oh, P.S. Don't forget the $1$ that adds to the comb. – Mark Viola Aug 03 '16 at 20:23

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