I believe f(x) = 0 is the only polynomial function that is both even and odd. I don't know about the other functions (exponential, logarithmic, logistic, trigonometric,...).
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Forget (for the moment) that we are talking about functions and just consider the value y of f(x) at the point x.
As f is both odd and even y=f(x)=f(-x) and odd y=f(x)=-f(-x) and so:
y=-y.
Now what number/s (real or complex) is equal to minus times itself?
But this is true for every value taken by f.
CB