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renjean
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thanks!
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renjean said:Homework Statement
Homework Equations
Give an example of a function f that is uniformly continuous on [-1,1] such that
sup{ [f(x)-f(y) / [x-y] } = infinity
The Attempt at a Solution
I have tried to come up with functions for hours but I am just not getting it. Any help would be appreciated.
chairbear said:I thought that any continuous function on a closed and bounded interval is also uniformly continuous.
And in the case of x*sin(1/x) the derivative appears to go to infinity at x=0.
chairbear said:I feel silly for making things more complicated than necessary. What was the easier example you had in mind? I was thinking square root x would work if the interval was [0,1].
chairbear said:Thank you for your help. I was wondering if you could help me to get started on another question I have.
I have to prove that for a function f with f'(0)=0, there's a sequence xn that converges to 0 for all n such that f'(xn) converges to 0. and xn can't = 0 for any n.
I'm not sure exactly how to get started on this, because I'm not sure if it's supposed to be a rigorous proof, or if I'm just supposed to come up with a sequence that satisfies the conditions for some f.
chairbear said:Sorry, there's a condition also that f: R-->R and must be differentiable on R
I like Serena said:Just out of curiosity, renjean and chairbear, what is the reason you deleted your questions?
Uniform continuity is a mathematical concept that describes the behavior of a function. A function is considered to be uniformly continuous if, for any given value of the input variable, the output of the function does not change by a large amount when the input is changed by a small amount. In other words, the function's behavior is consistent and predictable.
Uniform continuity differs from regular continuity in that it considers the behavior of a function over the entire domain, rather than just at a single point. Regular continuity only requires that the function be continuous at a specific point, whereas uniform continuity requires the function to be continuous across the entire domain.
The supremum of a function is the least upper bound of the range of the function. In other words, it is the smallest value that the function can take on, without ever exceeding it. In terms of graphing a function, the supremum is the highest point on the graph.
The concept of supremum is important in mathematics because it allows us to define and compare the behavior of functions, especially in cases where the functions may not have a maximum or minimum value. It also helps us to prove the existence of certain mathematical objects, such as limits and derivatives.
Uniform continuity and supremum are related in that uniform continuity is often used to prove the existence of the supremum of a function. In other words, if a function is uniformly continuous, we can use that information to show that the supremum of the function exists and is a well-defined value.