Delta functions live under integral signs, and two expressions (D1(x) and D2(x)) involving...

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.)

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(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).