Uniform Continuity: Definition & Applications

[saminny]

Hi,
This may sound lame but I am not able to get the definition of uniform continuous functions past my head.

by definition:
A function f with domain D is called uniformly continuous on the domain D if for any eta > 0 there exists a delta > 0 such that: if s, t D and | s - t | < delta then | f(s) - f(t) | < eta. Click here for a graphical explanation.

I can just choose "delta" that is a large number that will make any 2 points on the curve satisfy this condition. 1/x would be uniform continuous if I simply choose a large enough delta.

moreover, what is the utility and application of uniform continuous fynctions?

thanks,
Sam

 

[quasar987]

I think what you're missing is that in the definition, the part that says

"if s, t D and | s - t | < delta then | f(s) - f(t) | < eta"

actually means

"if for all s,t in D such that |s-t|<delta, we have | f(s) - f(t) | < eta."

So it does not suffice that you can find two points of D a distance less than delta apart that satisfy | f(s) - f(t) | < eta, but rather, the definition is saying that all the points in D that are a distance less than delta apart must satisfy | f(s) - f(t) | < eta !

 

[daniel_i_l]

Hi,
moreover, what is the utility and application of uniform continuous fynctions?

For starters, in order to prove that the Darboux integral is defined for any continuous function on a closed interval, Cantors theorem - which states that a continuous function on a closed interval is uniformly continuous - is used and then the uniform continuity is used to prove that the integral is defined.