Q:

Dirac delta equality

Does the equality below hold?
$$f(x)=\int_{-\infty}^{\infty}f(x')\delta(x-x')dx=\int_{-\infty}^{\infty}f(x')\delta(x'-x)dx$$

A:

Well,
$$\int_{\mathbb R} f(x)\delta (x - a)dx = {f(a)\int_{- \infty }^\infty {\delta (x - a)dx}} = f(a) = {f(a)\int_{- \infty }^\infty {\delta (a - x)dx}}$$
so YES, if you change your $dx$ to $dx'$.

