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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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