# What are functions called that yield the same result when applied to one of their result values

Suppose one of the following holds for all x, or all y, or all z:

f(f(x)) = f(x)
f(f(y)) = f(y)
f(f(f(z))) = f(f(z)) = f(z)


... how do you call such a function?

• If the property existed only for a given set of inputs, I would most likely state it as "$f(x)$, $f(y)$, and $f(z)$ are fixed points of $f$" or, with a different wording, "$f(x)$, $f(y)$, and $f(z)$ are fixed by $f$". If $f$ is idempotent, then $f(x)$ is fixed by $f$ for all $x$. Feb 1 '16 at 12:13
• I thought about "fixpoint", but those are $x$ with $f(x) = x$ -- not the same as $f(f(x)) = f(x)$. I don't know a term for "partially idempotent" functions. (Of course, if $f$ is some algorithm we iterate, e.g. a numerical one, then we'd say the final result -- when $f^{(n)}(x) = f^{(n-1)}(x)$ -- is a "fixpoint".)