Terminology for "support"-like word meaning subset of a function's domain not mapped to its argument, i.e., $f(x)\neq x$, not the identity.

Question

As per title, is there a word/terminology for the subset of a function's domain that it doesn't map to its argument, i.e., for $f:\mathbb{R\to R}$, define $\mbox{not_identity}(f)=\left\{x\in\mathbb{R}\middle|f(x)\neq x\right\}$. Is there specific terminology for "not_identity()"?

score 2 · Accepted Answer · answered Oct 12 '22 at 12:25

2

I have never seen a term for such a point. The compliment is known as the set of fixed points of $f$. You could say "support of $f(x)-x$" or you could say "set of non-fixed points of $f$" or define a new term if those are two clunky. In some applications, "fixed points" are called "invariant points" in which case it might make sense to call non-fixed points "variant points" though you shouldn't use that without first defining it.

answered Oct 12 '22 at 12:25

Dark Malthorp

5,945

Thanks, DM. I was definitely aware of fixed points (have an ms in physics). And, yeah, okay, if there's no specific terminology then I guess your "support of $f(x)−x$" is probably as good as it gets. But functionally, I think it's maybe a little better to refer to it as $\mbox{support}(f-id)$, where $id(x)=x$ denotes the identity. That hadn't crossed my mind, though now that you suggest it, seems pretty obvious. Thanks for mentioning it. – John Forkosh Oct 12 '22 at 16:22

Terminology for "support"-like word meaning subset of a function's domain not mapped to its argument, i.e., $f(x)\neq x$, not the identity.

1 Answers1