About single-valued function 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?

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)
