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.