What makes a function not defined? I have started studying precalculus and would then start up with calculus. While studying about functions I wondered whether this function would be defined at $a$ or not. Take a look at it. $$ f(x) = \frac{(x-a)(x-b)(x-c)...(x-n)}{(x-a)} $$
Here if we will simplify it further then the term $\left( x-a\right)$ would cancel out making the function defined at $a$ but if we would leave it as such it would be undefined at that point.
I asked this question because I found in some sources that the graph of such functions have an open dot at that point indicating it discontinuous at that point. But I couldn't explain it. Are the expressions before and after cancelling different or it's something else?
I would be highly obliged for your help and thanks ...

A function $f$ is a special asymmetric relation between two sets $A$ and $B$, represented by $f:A\to B$. The relation consist in that for every element of $A$ (called the domain of $f$) exists a unique element in $B$ (called the codomain of $f$) defined by the function $f$.
In your example, if the codomain of $f$ is $\Bbb R$, then $f(a)\notin \Bbb R$ because the division by zero is not defined in $\Bbb R$.
