Is there a word for *never* idempotent?

Question

I understand that a function is consider idempotent if $f(x) = f(f(x))$ for all $x$.

Is there a word for functions that are never idempotent for any possible input?

Answer 1

Let $f:S\to S$ map a set $S$ into itself.

The terminology of $f(x) = f(f(x))$ being idempotent when this hold for all $x\in S$ is okay, but the notion that it holds for some $x$ is a different concept.

We say that function $f$ has a fixed point when $f(x) = f(f(x))$ for some particular input $x$. The specific definition is that $x$ is a fixed point of $f$ iff $x = f(x)$, but if $f(x) = f(f(x))$ holds, then $f(x)$ is a fixed point of $f$ (and conversely, $x=f(x)$ implies $f(x) = f(f(x))$). Thus it is the same as saying a fixed point exists.

Functions that have no solution $f(x) = f(f(x))$ can then be described as fixed point free. In the case of automorphic (one-to-one) functions, a function without fixed points is often called a derangement.