I'm dealing with a class of continuous functions $f:[0, 1] \to \mathbb{R}$ such that the preimage of any $y \in f([0, 1])$ is of finite cardinality. I wonder whether there is a common terminology of such functions. Thanks!
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2A function with finite fibers? – azif00 Nov 24 '20 at 19:25
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2If the cardinality of the preimages is constant (that is, if there exists $n \in \mathbb{N}$ such that $|f^{-1}({y})| = n$ for all $y \in [0, 1]$) I have heard the term $n$-to-1, though not very commonly. – Duncan Ramage Nov 24 '20 at 19:27
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@azif00 Thanks for the reply. I wonder whether there is a specific name for functions with finite fibers. For example, in differential geometry, we call curves with non-zero derivatives as `regular curve'. – potionowner Nov 24 '20 at 19:28
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Related: A function that crosses each horizontal line only finitely many times. In classical real analysis the term Banach indicatrix (of a given function, at a given point) refers to the cardinality of the inverse image under that function of the singleton consisting of that point (see here and here), (continued) – Dave L. Renfro Nov 24 '20 at 19:48
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so one could refer to this as a function with a finite Banach indicatrix at each point. However, this term might not be sufficiently well known for general usage. – Dave L. Renfro Nov 24 '20 at 19:48
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The term 'finite-to-one' is a thing. – Tyrone Nov 24 '20 at 20:07
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I’ve seen the term “finite-to-one” being used in topology. No reference that I can recall.
Henno Brandsma
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