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Homework Help: Difference between Uniformally Continuous and Continuous

  1. Mar 22, 2012 #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. Mar 22, 2012 #2
    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. Mar 22, 2012 #3
    All uniformly continuous functions are continuous, but not all continuous functions are uniformly so. Here's a short PDF on the distinction:

  5. Mar 22, 2012 #4
    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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