# Uniform Continuity: Definition & Applications

• saminny
In summary, uniform continuous functions are defined as functions that satisfy the condition that all points within a certain distance from each other have a corresponding difference in their function values that is smaller than a predetermined value. This is useful for proving the existence of certain integrals, such as the Darboux integral.
saminny
Hi,
This may sound lame but I am not able to get the definition of uniform continuous functions past my head.

by definition:
A function f with domain D is called uniformly continuous on the domain D if for any eta > 0 there exists a delta > 0 such that: if s, t D and | s - t | < delta then | f(s) - f(t) | < eta. Click here for a graphical explanation.

I can just choose "delta" that is a large number that will make any 2 points on the curve satisfy this condition. 1/x would be uniform continuous if I simply choose a large enough delta.

moreover, what is the utility and application of uniform continuous fynctions?

thanks,
Sam

I think what you're missing is that in the definition, the part that says

"if s, t D and | s - t | < delta then | f(s) - f(t) | < eta"

actually means

"if for all s,t in D such that |s-t|<delta, we have | f(s) - f(t) | < eta."

So it does not suffice that you can find two points of D a distance less than delta apart that satisfy | f(s) - f(t) | < eta, but rather, the definition is saying that all the points in D that are a distance less than delta apart must satisfy | f(s) - f(t) | < eta !

saminny said:
Hi,
moreover, what is the utility and application of uniform continuous fynctions?
For starters, in order to prove that the Darboux integral is defined for any continuous function on a closed interval, Cantors theorem - which states that a continuous function on a closed interval is uniformly continuous - is used and then the uniform continuity is used to prove that the integral is defined.

## 1. What is the definition of uniform continuity?

Uniform continuity is a mathematical concept that describes the behavior of a function when its input and output values are close to each other. A function is uniformly continuous if for any small change in the input, there is a corresponding small change in the output.

## 2. How is uniform continuity different from continuity?

Uniform continuity is a stronger condition than continuity. While a continuous function only requires small changes in input to result in small changes in output at a specific point, uniform continuity requires this behavior to hold for the entire domain of the function.

## 3. What are the applications of uniform continuity?

Uniform continuity has many applications in mathematics and in other fields, such as physics and engineering. It is used to prove the existence and uniqueness of solutions to differential equations, to study the convergence of series and integrals, and to analyze the behavior of functions in optimization problems.

## 4. How can uniform continuity be tested?

To test for uniform continuity, one can use the definition of the concept directly, by checking if for any given small change in the input, there exists a corresponding small change in the output. Another method is to use the theorem that states that a function is uniformly continuous if and only if it is continuous and its derivative is bounded on the entire domain.

## 5. Can a function be uniformly continuous on one interval but not on another?

Yes, a function can be uniformly continuous on one interval but not on another. This is because the behavior of a function can vary across different intervals. A function can satisfy the conditions of uniform continuity on one interval but fail to do so on another due to the difference in the behavior of the function on those intervals.

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