algebra precalculus - 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 ...

Answer

Consider the following functions: $$f(x) = 5$$

$$g(x) = \dfrac{5(x-2)}{(x-2)}$$

The function $g$ is not defined at $x = 2$, but agrees with $f$ at every other point.

So we would say these functions are not the same, because their domains are different.