Intuition for the epsilon-delta definition of continuity This is my first question so I hope this sort of thing is OK to ask. I'm working my way through Rudin's Principles of Mathematical analysis, and I'm up to chapter 4, which is on continuity in the context of functions between metric spaces. It introduces what I understand to be the standard epsilon-delta definition used in calculus, but I'm struggling to gain an intuitive understanding of what it means. I came up with what I think is an English version of the gist of it:

A function f is continuous at some point p in its domain iff
  sufficiently small deviations from p result in arbitrarily small
  variations in f(p).

Does this show the general idea of continuity? If not, how should it be changed to fix it? Thanks in advance for any answers :)

Rudin's a tough one. Check out this Khan Accademy video on the topic of limits: https://www.youtube.com/watch?v=-ejyeII0i5c
The notion of a limit captures entirely the definition of continuity, which explains the equivalence of limits and continuity gotten later in the chapter (A function is continuous at $x$ if the limit can move in and out of the function for any sequence converging to $X$.).
