We know that the if a function $f$ is continuous in a point $a$ if and only if $\forall \epsilon>0, \exists \delta >0$ such that if $$|x-a|<\delta\implies|f(x)-f(a)|<\epsilon$$ But note that in this definition we already assume that $f(a)$ exists, in other words the function is well defined in the point $a$, which mean that the definition works only with jump discontinuity and infinite discontinuity .
This is what i understand from this definition.
On
I think you are confused with the definition of limit and continuity. While we can talk about the limit of a function at a point where it isn't defined (as long as this point is an accumulation point of the domain) we can't talk about continuity for a point that doesn't belong in the domain of the function.
In words: a function is continuous at a point if it is defined at that point and it has a limit at that point that agrees with its value.
That's just what the formal definition says.
If a function is not defined at a point it can't be continuous there, so this definition is correct when the graph has a vertical asymptote so is not defined.
It's also correct at what looks like a jump discontinuity. If the function is defined at the jump point its value can't be the limit there.