9
In mathematics we would say $f$ is an idempotent function. It's a widely known term and I suppose most TCS people should also recognize it.
1
Ilkka Törmä has already given the answer that $f$ is an idempotent function.
You might want to be aware of the concept of
fixed-points.
A point $c$ is a fixed-point for $f$ if $f(c) = c$, and hence
$$f(f(c)) = f(c) = c.$$
A set of fixed-points is often called a fixed set.
In your scenario, the output of $f$, or
image of $f$, is
a fixed set itself.
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