6

This is my first question so I hope this sort of thing is okay 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 :)

J. W. Tanner
  • 60,406
Escadara
  • 160
  • 1
    This is the right intuition for the concept of continuity... if you're thinking about functions reals->reals or between any other "continuous" sets. But on discrete metric spaces, like N or Z, it doesn't really work: the definition is still meaningful, but it doesn't feel the same way, as you can't make small deviations from a point -- you've got to move by at least one or not change the point at all. So my opinion is: yes, this is the right intuition where appropriate, and it's very important too; but be prepared to abandon it when dealing with some other metric spaces. – zipirovich Aug 31 '16 at 05:55
  • 1
    I'd really like to add back the [tag:metric-spaces] tag, but all of the 5 tags I have now seem more important. – Caleb Stanford Aug 31 '16 at 06:19
  • 1
    start with an easier book e.g. "yet another introduction to analysis' by Bryant – Mark Joshi Aug 31 '16 at 06:48
  • 1
    @zipirovich thanks, I think I get what you're saying, in that this intuition only works for limit points of the domain (ie only works for the entire domain when its dense-in-itself). As I understand it every function defined on a space like N or Z is continuous (eg delta=0.5 works for all epsilon), so how would that be intuitively explained? Thanks again :) – Escadara Aug 31 '16 at 07:13
  • @MarkJoshi thanks for the advice :) I hadn't heard of that book before but the reviews for it seem pretty good. I've been slowly working through Rudin for the last 6 months or so, and it's pretty slow going, so maybe it would be a good idea to use it as a supplement for when I'm stuck. – Escadara Aug 31 '16 at 07:13
  • 1
    see my discussion at http://www.markjoshi.com/RecommendedBooks.html#analysis – Mark Joshi Aug 31 '16 at 23:31
  • 1
    @Escadara: Yes, that's true - any function is continuous at any isolated point of the domain. How to explain this intuitively? I'm not sure; I guess it's a matter of opinion. Here's one way to look at it. Let's think of functions on the real line. If $f(x)=x^2$ on a domain like $[1,2]\cup[3,4]$, then the graph consists of two disjoint parts, but it feels reasonable to call each part continuous, i.e. $f(x)$ is continuous on $[1,2]$ and on $[3,4]$ - each part is a continuous curve. But by the same token, the part of the graph that corresponds to an isolated point is continuous too. – zipirovich Sep 01 '16 at 00:15

3 Answers3

2

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.

Arthur
  • 199,419
  • Basically $p$ which is the point of continuity for our function $f$ variations on that functions value remain constant ? – Zophikel May 25 '17 at 02:28
  • @Zophikel Yes, if we're checking continuity at a point, then that point remains in focus the whole time. If we want the function to be just continuous, then that means by definition that it is continuous at all points. – Arthur May 25 '17 at 04:11
1

Rudin's a tough one. Check out this Khan Academy 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 $X$.).

J. W. Tanner
  • 60,406
user357980
  • 1,854
1

That is a very good intuitive description of continuity of a function between metric spaces. There are many equivalent definitions of continuity of a function. It is often useful to use one rather than another :For example: For metric spaces $(X,d)$ and $(Y,e),$ a function $f:X\to Y$ is continuous

iff (1) Whenever $x\in X$ and $(x_n)_{n\in N}$ is a sequence of members of $X,$ then $\lim_{n\to \infty}d(x,x_n)=0\implies \lim_{n\to \infty}e(f(x),f(x_n))=0$

iff (2) $\{x\in X:f(x)\in U\}$ is an open subset of $X$ whenever $U$ is an open subset of $Y$

iff (3) $\{x\in X: f(x)\in V\}$ is a closed subset of $X$ whenever $V$ is a closed subset of $Y.$

DanielWainfleet
  • 57,985