Even functions, odd functions, and the zero function???

Question

So my math teacher mentioned in class the other day in passing that if function f is both even AND odd then it must bue the zero function. He didn't really explain it though and it doesn't make a ton of sense to me. Is this true? If it is, can you provide a short proof that explains why it works for ALL functions f and not just a few examples?

Answer 1

Suppose f is both even and odd. Then for all x, f(-x) = f(x) and f(-x) = -f(x), so by transitivity -f(x) = f(x), so 2f(x) = 0, so f(x) = 0. Since this holds for all x, f is the zero function. Q.E.D.

Answer 2

An even function when added to another even function gives an even function. The sum of an even and odd function cannot be categorized as even or odd, eg f(x) = x, is an odd function. f(x) = x squares, is an even function. F(x) = x + x(squared) is neither even not odd. 2) The product of an even function and an odd function is an odd function. The product of two odd functions is an even function. The product of two even functions is an even function.