Why are these two delta function equal

Question

In my system textbook it claims that

$$\delta(x)=\delta(-x)$$

I understand the proof as follow

$$\int_{-\infty}^\infty f(x)\delta(-x)\,dx$$

let $u=-x\,\:,\: du=-dx$

$$\int_{-\infty}^\infty f(x)\delta(-x)\,dx=\int_{-\infty}^\infty -f(-u)\delta(u)\,du =\int_\infty^{-\infty} f(-u)\delta(u)\,du=-f(-0)=-f(0)\neq f(0)$$

how do I resolve the negative sign in the end, the textbook entirely disregard the sign issue

any hint would be much appreciated

Answer 1

Your first equality isn't correct; making the change of variable $t = -x$ actually gives

$$\int_{-\infty}^{\infty} f(x) \delta(-x) dx = \int_{\infty}^{-\infty} f(-t) \delta(t) d(-t) = \int_{-\infty}^{\infty} f(-t) \delta(t) dt$$

Answer 2

This comes from the scaling property of the dirac delta function: $$\delta(ax) = \frac{1}{|a|}\delta(x). $$ So you would have $$\delta((-1)x) = \frac{1}{|-1|}\delta(x) = \delta(x)$$