Definition of an even or odd function

Question

I am a little confused about the definition of an odd function and an even function. The additive inverse part is clear but my question is does the function have to be univariate? Does it have to be from $\mathbb{R} \to \mathbb{R}$? Any help in this direction would be helpful. Thanks.

Answer 1

With multivariate functions, oddness/eveness is general talked about in respect to a specific variable. $f(x,y)$ is odd with respect to $x$.

Answer 2

Well, with multivariable functions there is a generalisation of parity. So, a single variable function, call it f(x), is even if f(x)=f(-x) for every value of x in the domain. f(x) is called even if f(x)=-f(-x). Now, with multivariable functions there is a generalisation of parity, as I said at the start. Here though functions aren't called even or odd: they are called even symmetric or odd symmetric. A multivariable function, call it f(x_1, x_2,x_3,...,x_n) (where x_n is x sub n), is even symmetric if f(x_1, x_2,x_3,...,x_n)=f(-x_1, -x_2,x_3,...,-x_n). f(x_1, x_2,x_3,...,x_n) is called odd symmetric if f(x_1, x_2,x_3,...,x_n)f(-x_1, -x_2,x_3,...,-x_n). As in the single variable world, the only function that is odd and even is f(x_1, x_2,x_3,...,x_n)=0. As in the single variable world, there are functions which are neither odd nor even.