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?
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?
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.