Difference between Uniformally Continuous and Continuous

[kingstrick]

I don't see the subtle differences between continuous and uniformally continuous functions. What can continuous functinons do that unifiormally continuous functions can't or vice versa?

 

[zooxanthellae]

For continuous functions, if we set a point x1 and some E>0 then I can give you a d>0 st. for |x-x1| < d then |f(x) - f(x1)| < E.

For uniformly continuous functions, all I need is an E and I can give a d that works for ANY point x where the function is uniformly continuous. In other words, if two points - just random points - are within d of each other, then there f-values are within E of each other.

The difference is that in the first one, for a given E, d varies depending on what point your looking at. It may be that there is no d>0 that will work for all points. But for a uniformly continuous function, for any E>0 there is some d>0 that works for all points (where f is uniformly continuous).

 

[QuarkCharmer]

All uniformly continuous functions are continuous, but not all continuous functions are uniformly so. Here's a short PDF on the distinction:

http://www.math.wisc.edu/~robbin/521dir/cont.pdf

 

[Robert1986]

OK, so let's say that I claim that $f$ is continuous on some interval, $I$, let's say the open unit interval. Then if you give me an $x_0 \in I$ and an $\epsilon > 0$ I should be able to give you a $\delta$ such that $|f(x_0) - f(x)| < \epsilon$ whenever $|x_0 - x| < \epsilon$. Now, it is important to note that I get to "see" $\epsilon$ AND $x_0$ before I have to come up with $\delta$.

Now, if I claim that $f$ is uniformly continuous on some interval, $I$, say, the closed unit interval, then given an $\epsilon > 0$ I must come up with a $/delta$ that will work FOR ALL $x \in I$. That is, I don't get to "see" $x_0$ before I come up with $\delta$.

So, why is this important? Well, if $I$ is compact and $f$ is continuous on $I$ then $f$ is uniformly continuous on $I$. Many of the theorems about derivatives, integrals, approximation of functions, etc, in analysis (at least at the undergrad level, which is all I know) require that the function be continuous on some closed and bounded (and hence compact) interval. This uniform continuity can then used to prove whatever is being proven.