I would like to calculate $\int_{a}^{d}\delta(x-b)\delta(x-c)f(x)dx$, where $a \le b \le d$ and $a \le c \le d$.
My imediate thought is to integrate by parts to obtain $$\int_{a}^{d}\delta(x-b)\delta(x-c)f(x)dx = [\delta(b-c)f(x)]_{a}^{d} - \int_{a}^{d}\delta(b-c)f'(x)dx$$ $$= \delta(b-c)([f(x)]_{a}^{d} - [f(x)]_{a}^{d}) = 0$$
However, this does not seem correct. Any help on the matter would be appreciated.
So I believe the answer can be found using test functions to represent the $\delta$-function. I obtained the general form
$$ \int_a^d \delta(x - b) \delta (x - c) f(x) dx = \delta(b - c) f(c), $$
where $f$ is evaluated at $c$ w.l.o.g as the integral is 0 when $b \ne c$.