Let D be a nonemty set which isnot closed. Find a function f: D->R such that f is continuous on D but not uniformly continuous on D.

For this would X^2 be a function and how would I actually prove this?

- April 18th 2011, 07:43 AMalice8675309How I find a Function f that is continuous but not uniformly continuous
Let D be a nonemty set which is

*not closed*. Find a function f: D->R such that f is continuous on D but not uniformly continuous on D.

For this would X^2 be a function and how would I actually prove this? - April 18th 2011, 07:52 AMFernandoRevilla
- April 18th 2011, 08:12 AMalice8675309
- April 18th 2011, 08:24 AMFernandoRevilla

It is continuous (why?) . You can prove that is is not uniformly continuous choosing**epsilon = 10**. If there exists**0 < delta < 1**satisfyng the definition, take

**x = delta , y = delta / 11**

Then,

**| x - y | = ( 10 / 11 ) delta < delta**

and

**| f( x ) - f ( y ) | = ... = ( 10 / delta ) > 10**

which contradicts the definition.