Describe the distributional derivative of $f$

Question

Let $f$ be a piece wise defined function with piece wise continuous derivative. Describe the distributional derivative of $f$.

My try:

If I suppose that the jump discontinuities are at the points $-1$,$0$ and $1$. Then I will look to generalize it. Then I have $$\left(\frac{d}{dx}f,\phi\right)=-\left(f,\frac{d\phi}{dx}\right)=-\left(\int_{-a}^{-1}f\frac{d\phi}{dx}+\int_{-1}^{0}f\frac{d\phi}{dx}+\int_{0}^{1}f\frac{d\phi}{dx}+\int_{1}^{a}f\frac{d\phi}{dx}\right)$$ where $Supp \phi=[-a,a]$.

It will eventually come out to be $(f',\phi)$. Intuitively I think there has to be delta function somewhere.

Thanks for the help!!

Answer 1

Hint: Integrating by parts, $$\int_c^d f \dfrac{d\phi}{dx} = f(d-) \phi(d) - f(c+) \phi(c) - \int_c^d \dfrac{df}{dx} \phi$$