What is intution behind sequential definition of continuity and its relation with ordinary definition of continuity I was reading the sequential definition of continuity and it didnot made intiutively much sense to me and how it is related to ordinary continuity. Can someone please explain that to me.
Note- I am in undergrad course and and not yet exposed to Topology
Thanks
 A: I'll give you an explanation that some colleagues dislike. Sequential continuity (at a point $p$) means that no matter how you approach $p$ with a discrete motion (i.e. $x_n \to p$), the corresponding values $f(x_n)$ approach precisely $f(p)$. For this reason I visualize sequential continuity as a "dynamical" definition in some sense opposed to the "static" definition with $\varepsilon$ and $\delta$. For real-valued functions defined on $\mathbb{R}$ with the standard distance the two definitions coincide. The reason is topological, and I will not try to explain this since you do not know topology yet. 
As I said at the beginning, many colleagues believe that sequential continuity is as "stati" as the other definition, since nothing moves. This is true from the viewpoint of a topologist (by the way, continuity is always equivalent to another "dynamical" definition in terms of Moore-Smith sequences or nets, while sequential continuity can be weaker than continuity itself), but after all a discrete dynamical system is essentially a sequence that describes a discrete motion.
