Continuous Functions Homework: Examples & Justification

[HappyN]

Homework Statement

Find an example of a continuous function f:R->R with the following property.
For every epsilon >0 there exists a delta >0 such that |f(x)-f(y)| <epsilon whenever x,y e R with |x-y|<delta.
Now find an example of a continuous function f:R->R for which this property does nto hold.
Justify your examples carefully.

The Attempt at a Solution

I think the property they've stated is just the definition for continuity? so for the first part sin(x) would work?
but for the second part how can you find a function which is continuous where the definition of continuity doesn't hold?
if the property above differs from the definition of continuity please explain how? thanks

 

[Jory]

I think this definition stated here looks a bit like Uniform continuity, which has a subtle but important difference.

In 'normal' continuity, delta can depend on x, so f(x) : R -> R, f(x) = x^2, is continuous

With Uniform continuity, delta cannot depend on x, so is a global property of a function, not a pointwise property. f(x) : R -> R, f(x) = x^2, is not uniformly continuous. You can think of it as saying |f(x) - f(y)| has no upper bound for a given delta

Edit: You'll probably want to look at your definitions closely, as they look very similar