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I was reading a paper about compression algorithm:

In order to optimality fit the line segments to the curve, Bellman's algorithm assumes that the input data is a valid (i.e., single-valued) function; thus, the trajectory cannot contain no loops.

What does it means with valid function? I suppose he intends to have a single-valued function, but: what's a single-valued function? According to wikipedia definition:

A single-valued function is an emphatic term for a mathematical function in the usual sense. That is, each element of the function's domain maps to a single, well-defined element of its range.

But this sounds to me like an injective function. Can you confirm that both (single-valued function and injective function) mean the same thing?

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Of course not! A map $f \colon X \to Y$ is injective when $f(x_1)=f(x_2)$ implies $x_1=x_2$. But, strictly speaking, you have to know what a function is, i.e. what a single-valued function is.

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thanks. So what's the best way to translate single-valued in italian language? – Mariano May 4 '12 at 9:39
Probably "funzione univoca" or "funzione ad un solo valore". But I believe that multi-valued functions need a name, since functions already have one :-) – Siminore May 4 '12 at 9:52

They are not the same thing. In an injective function each element of the co-domain is mapped to by at most 1 element of the domain (so two distinct elements of the domain cannot be mapped to the same element of the co-domain). In a single-valued mapping (or a function) it is the other way round: each element of the domain is mapped to at most one element of the co-domain.

For example, the square root mapping (on positive real numbers) is not single-valued as the square root of 4 (for example) is not uniquely determined. The square root mapping maps 4 to both 2 and -2 and so is not single-valued. However, if we restrict the square root mapping and demand that the output be positive we get a function (an injective function as it turns out)

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