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 :)

That is an almost correct intuitive formulation of what continuity is. Somehow you also need to get across that the actual size of the allowable deviations does not have anything to do with it. You could do that by saying "for any interpretation of the word 'small'", or something like that.
It does definitely show the general idea, though. Just for completeness, the general idea is formalised by the $\epsilon$-$\delta$ definition:

A function $f$ is continuous at a point $p$ in its domain if, for any $\epsilon > 0$, there is a $\delta > 0$ such that 
  $$
|x-p| < \delta \implies |f(x) - f(p)| < \epsilon
$$

The translation is that $\epsilon$ is the given bound on allowable variations in function value. $\delta$ is the bound you find on deviations from $p$ that keeps the function value within the given $\epsilon$-bound.
