Delta functions live under integral signs, and two expressions (D₁(x) and D₂ (x)) involving delta functions are said to be equal if

for every (ordinary) function f(x).
(a) Show that

where c is a real constant. (Be sure to check the case where c is negative.)
(b) Let θ (x) be the step function:

(In the rare case where it actually matters, we define θ (0) to be 1/2.) Show that dθ/dx = δ(x).

+00 = 1+00 f (x) D (x) dx = f (x) D (x) dx,