Symmetric PDF proof

Question

If F(x_u) = u, then show that if f(-x) = f(x), then x_1-u=-x_u where:

F is CDF

f is PDF.

In my mind, I understand why this is the case, but I'm having trouble writing a proof to show it.

Thanks

Answer 1

Suppose $\int_{-\infty}^af(x)dx=u$, and define $I:=\int_{-\infty}^{-a}f(x)dx$. By $x\mapsto -x$, $I=\int_a^\infty f(-x)dx$. Since $f$ is symmetric (or even, if you prefer that term),$$I=\int_a^\infty f(x)dx=\int_{-\infty}^\infty f(x)dx-\int_{-\infty}^af(x)dx=1-u.$$