Question

Why is f(x)=0 the only odd and even function?

Answer 1

If f is even, then f(-x) = f(x) for all x in the domain
If f is odd, then f(-x) = -f(x) for all x in the domain

Use substitution to find that...
f(-x) = -f(x)
f(x) = -f(x) <<--- replaced f(-x) on the left side with f(x)
f(x)+f(x) = 0 <<--- add f(x) to both sides and isolate f(x)
2*f(x) = 0
f(x) = 0

So if x is both even and odd at the same time, then f(x) = 0 for all values of x in the domain of f. This produces a flat horizontal line through the origin. It is both symmetric with respect to the y axis and it has origin symmetry as well (we can rotate it 180 degrees around the origin to have it line up with itself)

Answer 2

f is odd if and only f f(-x) = -f(x) so f(x) = 0 works because 0 = 0 always.

f is even if and only if f(-x) = f(x) so f(x) = 0 works because there's no x term so nothing changes. 0 = 0.