Remember that the Dirac delta function is well-defined only after it is integrated over.

(a) Prove that $$\displaystyle{\delta(ax) = \frac{1}{|a|} \delta(x)}$$.

(b) Show that $$\displaystyle{u_c(x) = \int_{-\infty}^x \delta(x'-c)dx'.}$$

(c) Show that $$\displaystyle{\delta(x-c) = \frac{d}{dx}u_c(x)}$$.