jibbles said:surely you meant bounded instead of unbounded, right?
Office_Shredder said:Is your definition of unbounded weird? Because if E=R then it's saying that all continuous functions are uniformly continuous.
The example of Z isn't because Z is Z is unbounded, the key property is that Z is discrete - any discrete set has that all functions on them are uniformly continuous.
Bachelier said:Yea I agree. That is why I am asking. Please see the attached theorem 4.20. The assumption on the boundedness is at the end of page 2
lavinia said:the theorem says there exists a continuous function on E that is notuniformly continuous
He means there are some noncompact unbounded sets E for which all continuous functions on E are uniformly continuous. Of course, any unbounded set is noncompact, so he is saying that there are some unbounded sets E for which all continuous functions on E are uniformly conitnuous.Bachelier said:Maybe I am reading too much into this.
After equation (23), Rudin writes:
"...Assertion (c) would be false if boundedness were omitted from the hypotheses."
Can you explain this further? especially via an example without using the set of integers.
Continuity is a mathematical concept that describes a function's behavior at a specific point or interval. A function is considered continuous if its graph is unbroken with no gaps or holes. This means that the value of the function at a given point is equal to the limit of the function as it approaches that point.
To determine if a function is continuous, you need to check three criteria: that the function is defined at the point in question, that the limit of the function as it approaches the point exists, and that the limit is equal to the value of the function at that point. If all three criteria are met, the function is continuous at that point.
Continuity and differentiability are related concepts, but they are not the same. Continuity is the property of a function being unbroken at a point, while differentiability refers to the smoothness of a function at a point. A function can be continuous without being differentiable, but if a function is differentiable, it must also be continuous.
Yes, a function can be continuous but not differentiable. This can happen when a function has a sharp corner or a cusp at a point. The function is still continuous because it is unbroken, but it is not differentiable because it is not smooth at that point.
To prove that a function is continuous, you need to show that it meets all three criteria for continuity: that it is defined at the point in question, that the limit of the function as it approaches the point exists, and that the limit is equal to the value of the function at that point. This can be done using mathematical proofs or by graphing the function and visually demonstrating continuity at the point in question.