Uniform Continuity: Showing f*g Is Uniformly Continuous on Bounded X

In summary, if f and g are uniformly continuous and bounded on X, then their product, f*g, is also uniformly continuous. This can be shown by using epsilon-delta proofs and the fact that the product of bounded functions is also bounded.
  • #1
CarmineCortez
33
0

Homework Statement


suppose f and g are uniformly continuous functions on X

and f and g are bounded on X, show f*g is uniformly continuous.


The Attempt at a Solution



I know that if they are not bounded then they may not be uniformly continuous. ie x^2
and also if only one is bounded they are not necessarily uniformly continuous.

not sure what to do if they are both bounded
 
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  • #2
CarmineCortez said:

Homework Statement


suppose f and g are uniformly continuous functions on X

and f and g are bounded on X, show f*g is uniformly continuous.


The Attempt at a Solution



I know that if they are not bounded then they may not be uniformly continuous. ie x^2
and also if only one is bounded they are not necessarily uniformly continuous.
No, you don't "know" that. In fact, here, you are told that they are both uniformly continuous.

not sure what to do if they are both bounded
Are you saying you could do this if only one were bounded? Do you understand what it is you are asked to prove?
 
  • #3
uniform continuity is intuitively a bit like saying the function doesn't have an infinite slope anywhere...
And if they are both bounded, is their product too bounded?
 
  • #4
Get your epsilons and deltas out. You want to show |f(x)g(x)-f(y)g(y)| can be made uniformly small if |x-y| is small. Hint: |f(x)g(x)-f(y)g(y)|=|f(x)g(x)-f(x)g(y)+f(x)g(y)-f(y)g(y)|. |f(x)-f(y)| and |g(x)-g(y)| can be made small since they are uniformly continuous. Do you see why f and g need to be bounded? Use epsilons and deltas to make the meaning of 'small' precise.
 

What is uniform continuity?

Uniform continuity is a mathematical concept that describes a function's behavior when its input and output values are close together. It states that for any given small change in the input, there exists a small enough change in the output so that the function remains close to its original value.

How is uniform continuity different from regular continuity?

While regular continuity requires the function to be continuous at every point within its domain, uniform continuity only requires the function to be continuous within a bounded interval. This allows for a more relaxed definition of continuity and is useful when dealing with functions that may have abrupt changes or discontinuities.

What does it mean for f*g to be uniformly continuous on bounded X?

For f*g (the product of two functions) to be uniformly continuous on a bounded interval X, it means that the function remains close to its original value when the input values are close together. In other words, there exists a small enough change in the input that results in a small enough change in the output, regardless of where the input is within the bounded interval.

How is uniform continuity proven for f*g on bounded X?

To prove uniform continuity for f*g on a bounded interval X, we must show that for any given value of epsilon (ε), there exists a corresponding value of delta (δ) such that for all x and y within the bounded interval, if the distance between them is less than delta, then the distance between their corresponding outputs will be less than epsilon.

Why is uniform continuity important in mathematics?

Uniform continuity is important because it allows us to analyze the behavior of functions within a bounded interval, without having to consider every single point within that interval. This simplifies the analysis and makes it easier to prove the continuity of a function. It is also a fundamental concept in calculus and is used in many other areas of mathematics, including analysis, topology, and differential equations.

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