A student asked me to demonstrate why a specific function $f(x)$ was odd about a point $a$. I realized that I never had actually formalized, let alone proven such a claim, but had rather rode the results of my intuition. Anyway, I came up with this
$f(x)$ is odd about $a \iff f(-x+a) = -f(x+a)$
Similarly,
$f(x)$ is even about $a \iff f(-x+a) = f(x+a)$
Does this check out? It makes sense to me, as we're translating the function back to the origin, then checking for parity there. This is equivalent to declaring the parity about $a$.
Also, if the title or my terminology should be brushed up, please do tell. Thanks.
EDIT: To make this more of a Math Exchange question. How would one prove this?
You cannot prove a definition correct. You can only explain why you choose to define something in a certain way. Moreover, what do you want odd and even about a point to mean? If you want "even about $a$" to mean "remains the same when reflected about $x = a$", and "odd about $a$" to mean "remains the same when rotated 180 degrees about $(a,0)$, then your definitions are both correct.
If however you have already defined "odd" and "even" for functions without using the algebraic definition, then you'll have to state precisely how you did so, if not we can't prove what we don't know. Note that "reflection" and "rotation" are most easily defined algebraically, so it's going to amount to almost a tautology if you do it that way.