How can we show f is periodic

Show that, if f:R → R is symmetric wit respect to 2 distinct vertical axes, then f is periodic.

Thank you.

Let the axes be
x = a
x = b.

Then we have the following for every real x:
f(x) = f(2a-x)
f(x) = f(2b-x)

In particular, this shows:
f(x) = f(2a - x) = f(2b - (2a-x) ) = f(x + 2(b-a))
so that f is at least 2|b-a|-perioidic.