Checking Continuity in General. Suppose you have a given  Function on $\mathbb R^n$.And you want to check the continuity on a Point.How can you be sure that you checked all possible ways to approach that point$?$That method is useful for proving it does not exist but to prove it does exist you need a more general method.But all methods for continuity have "for all ..." in them so how can you check "for all..."  (sequences ,Open sets,approaches..) whatever definition you wanna use.
 A: First of all, all norms on $\Bbb R^{n}$ are equivalent (do you know what that means?).  As a consequence, if a function is continuous with respect to one norm, it is continuous with respect to every norm, so you can just pick the norm that makes the proof of continuity easiest based on the given function.
Also, just as we prove continuity in the $1$-dimensional case, we have to show for every $\epsilon > 0$, blah blah blah.
To prove it, we let $\epsilon >0$ be arbitrary, i.e., instead of checking each $\epsilon > 0$ on its own (which would mean us checking uncountably infinite points), we do all of the work at once.  We find a $\delta$ that is a formula based on $\epsilon$, so that given any $\epsilon$, you can compute a $\delta$ that works.  It's the exact same in multi-dimensions.  Let $\epsilon$ be arbitrary, and find a working $\delta$ which is some formula with $\epsilon$ in it.  Then for any $\epsilon$ you pick, the formula will give you a $\delta$ that works.
A: The concept that you are thinking of is sequence continuity (which is equivalent to continuity in metric spaces such as R^n but that requires proof).
If you check the condition "for every given open set containing the image there exists an open set (containing the original point) that is mapped into the given open set" then you have continuity by definition. An open set includes its points from all directions at once.
The typical procedure (in metric spaces) is to use open balls centered at the given original point or its image: for every positive epsilon there exists a positive delta such that the open ball B(p,delta) is mapped inside B(f(p),epsilon).
