Understanding classification of discontinuities $\forall a,b\in \mathbb{R}$ , let  $\displaystyle f\left(x\right)\:=\:\begin{pmatrix}\frac{x^2+ax+b}{x^2-8x+15} & x\ne 3,5 \\0 & x=3,5\end{pmatrix}$.  
I need to Find all the discontinuities and classify them.  
I'm not really sure what to do,  because it $\forall a,b\in \mathbb{R}$.
What can i say about $x=3,5$? 
 A: The discontinuities depend on $a$ and $b$.
If $x-3$ divides the numerator, $f(x)$ has a limit at $x=3$; otherwise, not. Similarly with $x-5$ and $x=5$. The function will be continuous if, in addition, the limit is zero. Note that if $x-3$ divides the numerator, the numerator is then $(x-3)(x-c)$ for some $c$, and you can easily see what $a$ and $b$ are then.
Now set conditions on $a$ and $b$ based on those statements. You will get up to four possibilities: $f(x)$ continuous everywhere except at $3$ and $5$, everywhere except $3$, everywhere except $5$, and everywhere. Check carefully, as some of these may not be actual possibilities.
Do you need more hints?
(Note: my original answer was incorrect.)

We can easily see that $f(x)$ is continuous where $x$ is neither $3$ nor $5$. A quotient of continuous functions is continuous except where the denominator is zero. Polynomials are continuous, so this applies to your $f(x)$, and if $x$ is neither $3$ nor $5$ the denominator is not zero (since the denominator is $(x-3)(x-5))$.
