Difference between Uniformally Continuous and Continuous

  1. I don't see the subtle differences between continous and uniformally continuous functions. What can continuous functinons do that unifiormally continuous functions cant or vice versa?
     
  2. jcsd
  3. 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).
     
  4. 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
     
  5. OK, so let's say that I claim that [itex]f[/itex] is continuous on some interval, [itex]I[/itex], let's say the open unit interval. Then if you give me an [itex]x_0 \in I[/itex] and an [itex]\epsilon > 0 [/itex] I should be able to give you a [itex] \delta [/itex] such that [itex]|f(x_0) - f(x)| < \epsilon [/itex] whenever [itex]|x_0 - x| < \epsilon[/itex]. Now, it is important to note that I get to "see" [itex]\epsilon[/itex] AND [itex]x_0[/itex] before I have to come up with [itex]\delta[/itex].

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

    So, why is this important? Well, if [itex]I[/itex] is compact and [itex]f[/itex] is continuous on [itex]I[/itex] then [itex]f[/itex] is uniformly continuous on [itex]I[/itex]. 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.
     
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