Two functions f/g Uniform Continuity

In summary, uniform continuity for two functions f/g means that for every epsilon > 0, there exists a delta > 0 such that for all x,y in the domain of f and g, if the distance between x and y is less than delta, then the distance between f(x) and g(x) is less than epsilon. This is different from regular continuity, which only requires a delta to exist for each point in the domain. Some common techniques for proving uniform continuity include the mean value theorem, the Cauchy criterion, and the use of Lipschitz functions. It is possible for a function to be uniformly continuous on one interval but not on another, as uniform continuity depends on the behavior of the function in a specific interval
  • #1
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I was wondering if f and g are two uniformly continuous functions on a set such that g(x) is not zero is f/g uniformly continuous?


I have a feeling it is not but I can't seem to find a counter example.
 
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  • #2
The function 1/x on ]0,1] should probably be a good counterexample...
 
  • #3
But 1 is not considering a function f is it?
 
  • #4
Take f(x)=1, the constant function, and g(x)=x...
 
  • #5
If D is compact then f/g will be uniformly continuous on D right?
 
  • #6
Yes. That's why I took the interval ]0,1].
 

What is the definition of uniform continuity for two functions f/g?

Uniform continuity for two functions f/g means that for every epsilon > 0, there exists a delta > 0 such that for all x,y in the domain of f and g, if the distance between x and y is less than delta, then the distance between f(x) and g(x) is less than epsilon.

How is uniform continuity different from regular continuity for two functions f/g?

Uniform continuity requires that the same delta works for all points in the domain, while regular continuity only requires that a delta exists for each point.

What are some common techniques for proving uniform continuity for two functions f/g?

Some common techniques for proving uniform continuity include the use of the mean value theorem, the Cauchy criterion, and the use of Lipschitz functions.

Can a function be uniformly continuous for two functions f/g on one interval but not on another?

Yes, a function can be uniformly continuous on one interval but not on another. Uniform continuity depends on the behavior of the function in a specific interval, so it is possible for a function to have different uniform continuity on different intervals.

How does uniform continuity for two functions f/g relate to the concept of a limit?

Uniform continuity is closely related to the concept of a limit. Both concepts involve the behavior of a function as the input approaches a certain value. Uniform continuity guarantees that the function will not have any sudden jumps or breaks, which is similar to the idea of a limit being the value that a function approaches as the input gets closer and closer to a certain value.

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