Thread: How 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?

Originally Posted by alice8675309

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 example f : ( 0 , 1 ] -> IR , f ( x ) = 1 / x

Originally Posted by FernandoRevilla

For example f : ( 0 , 1 ] -> IR , f ( x ) = 1 / x

Oh I see. But how would I prove that it satisfies the conditions :/

Originally Posted by alice8675309

Oh I see. But how would I prove that it satisfies the conditions :/

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.