Vrftsjtryk

here one question that may look stupid. Why in general one insists on naming in a different way functions and set-valued functions just because one is single valued and the other is not? I mean, from topology, we define as function an object that maps one topological space $X$ into another one $Y$, we never require this mapping to be single-valued! Thanks!

I believe we have lifted some of your confusion in the comments, but let me illustrate how being "set-valued" and being "multivalued" have nothing in common. Let $A = {1}$ be a one-point set and let $B = {1,A,{A}}$. We are going to define four relations from $A$ to $B$. By definition these are subsets of the three-element set $A times B$, and $a in A$ is related to $b in B$ if $(a,b)$ belongs to the relation.

Let $R_1 = {(1,1)}$. This is just an ordinary function: the input $1$ has the unique output $1$. Moreover it isn't really "set-valued" if we consider $1$ to be an object that is somehow not really a set because we haven't specified what it contains.

Let $R_2 = {(1,A)}$. Again this is just an ordinary function: the input $1$ yields the output $A$ (a set). This is a "set-valued function".

Let $R_3 = {(1,1),(1,A)}$. This is a multivalued function because the input $1$ has precisely two outputs, namely $1$ and $A$. It isn't precisely set-valued because you don't consider $1$ to be a set.

Let $R_4 = {(1,A),(1,{A})}$. This is a "set-valued" multifunction.

Your question, as it stands, is confusing, but from your discussion with others in the comment sections, I think the problem is that you are yet to understand what the word function means in modern mathematics. In particular, I think you are imposing additional constraints on the usual definition, namely that you think a function must be one-to-one. But this is not necessary at all. Many functions are not injective (that's synonymous to one-to-one), and an example is the one you plucked from topology in your discussion in the comment under the OP.

So a function need not map different points to different images. There is absolutely no need for that in general. All that we require is that the same point must have no less and no more than one image.

In the language of sets, a function takes each point in a set (called the domain) to exactly one image point in a set (called the codomain). This does not mean two points in the domain cannot have the same image (we make no such restriction), so long as they both have just that one point as image.

I hope this clarifies some things. But I think you should read more about functions and really understand them. A definition means just what it says, no more or less.

PS. I thought I should give an example. Note that what we call a constant function is any function that takes all the points in the domain to the same one point in the codomain. This is a function because every point of the domain has precisely one image, even though in this case they all have the same image.