Understanding the notion of semifunction I read in a notes: A semi-function is a relation (not a function) with of the form $y^2=f(x)$. 
It seems that we can get more that one values for $f(x)$ for a single value of $x$. 
Could any-one please help me to understand this notion.
The link of the note is here.
 A: "Semi-function" is a rather seldom-encountered term, I think. Multivalued function or set-valued function are the more common ones. If there are some specific issues you don't understand from those presentations, you should probably ask a more pointed question. 
For the $ y^2 = f(x) $ example, it simply means that $ y = \pm \sqrt{f(x)} $, so there is a set of two values corresponding to every $y$... assuming $ f(x) $ is positive... which the paper doesn't even say. If $ f(x) $ is negative (for some $x$), then there will be no real value $y$ for that $x$, i.e. the empty set is associated with the corresponding $y$. I guess the "semi" term wants to suggest that there are only two (or perhaps zero-or-two) rather than more values for every y, but I'm not exactly sure what the author had in mind when he chose this seldom-used "semi-function" terminology instead of the more common one(s) that I mentioned in the previous paragraph. The paper you link to lacks a more general definition of "semi-function" beyond this  $ y^2 = f(x) $ so we can't be sure what it would mean in general in this author's mind.
A: The notes say $f(x)$ is a function, so we can get exactly one value for $f(x)$ with a single value of $x$. However, we can get two values of $y$ given a single value of $x$, since it could be plus or minus.
