If I have a hypothetical multivariable function $f(x_1, \cdots, x_n)$, is there a term for the special case where we constrain the input variables to be equal?
That is to say, let $x_1 = x_2 = \cdots =x_{n-1} = x_n$ in $f(x_1, \cdots, x_n)$. Not all functions can satisfy this constraint due to indeterminant forms, and possibly other problems, so for the purposes of this question let's assume we're restricting ourselves to talking about functions that will admit this restriction.
Sometimes it is worth adding a few words to distinguish similar concepts, so let me offer a few words. I'm not asking for the term "symmetric function", where permuting the inputs of the function doesn't change the output value. I'm not asking for a family of functions per se, but rather the term for restricting a given function (which may or may not be symmetric over its domain) to the case where its arguments are equal.