I'm trying to prove the following statement:

Prove that the function f(x)=1/x is continuous on (0,1] but it is not uniformly continuous on this interval.

Thanks a lot.

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- Nov 4th 2010, 01:29 AMAKTiltedUniform Continuity for 1/x
I'm trying to prove the following statement:

Prove that the function f(x)=1/x is continuous on (0,1] but it is not uniformly continuous on this interval.

Thanks a lot. - Nov 4th 2010, 06:20 AMTinyboss
For the "not uniformly continuous" part: do you know how to formally negate a logical proposition, i.e. exchange "for-all's" and "there-exists's" and reverse (in)equalities? Try that with the definition of uniform continuity.

- Nov 4th 2010, 07:04 AMDrexel28
I actually discuss this in my post here. In essence, (we'll speak less generally now) if were uniformly continuous, then we could extend it to some uniformly continuous map . But, this is just nonsense since we'd have to have that

- Nov 4th 2010, 07:08 AMAlso sprach Zarathustra
- Nov 4th 2010, 07:15 AMAlso sprach Zarathustra
Another much better solution:

Choose and for any .

, but .