continuity of a function, why does p have to be in the domain (This could be duplicate, I am very sorry if it is the case, I couldn't seem to find it anywhere.)
The definition I normally use for continuity of a function at $p$ (domain $E$) is:
'given $ε > 0$ there exists $δ > 0$ such that for all $x$ in $E$ with $|p - x| < δ$ then $|f (p) - f (x)| < ε$'.
I am wondering why does p have be in the domain? 
Comparing with the definition of limit of a function, p just needs to be a limiting point, which may or may not be in the domain. Why is it different for these two cases?
Is it something to do with the fact that $|p - x| =0$ is included for continuity but $|p-x|>0$ for limit of a function? Or should I just take it as a definition and not argue why it is necessary?

If $p$ is not in the domain of $f$, then the expression $|f(p)-f(x)|$ is meaningless, since $f(p)$ does not exist. 
Consequently, the entire expression $|f(p)-f(x)|<\epsilon$, and also the logical statement that contains this expression, i.e. the statement $$\forall x\in E\forall \epsilon > 0 \exists \delta > 0: |p-x|<\delta \implies |f(p)-f(x)|<\epsilon$$ is also meaningless.
The expressions that define continuity only make sense if we demand that $p\in E$.
