Vertical line test 
A vertical line crossing the x-axis at a point $a$ will meet the set in exactly one point $(a, b)$ if $f(a)$ is defined, and $f(a) = b$.

If the vertical line meets the set of points in two points then $f(a)$ is undefined? 

(Short answer)
No.  Rather, we conclude $f$ is a relation, not a function.
Response to comment:
A real function of one variable is really saying three things:


*

*It's a real mapping.  This means that the object in question is a mapping from $\mathbb{R}^n \to \mathbb{R}^n$.  In "normal talk," this means that you can put any real number in, and you get real numbers out.  The object in question doesn't accept complex numbers.

*It's a function.  This means that every input has exactly one output.  That is, you can't put in a number and be able to get out multiple numbers.

*It's of one variable.  This means that there is only one value used for each input.  An example of a multivariable function is $f(x, y, z) = xyz$.  This is a function of 3 variables.


Does this help?
