# Lipschitz Continuity & Uniform Continuity: Showing sinx & cosx in R

• barksdalemc
In summary, to show that Lipschitz continuity implies uniform continuity, one needs to modify the proof for ordinary continuity. This is because Lipschitz continuity is a global property of a function, while uniform continuity is the global version of local continuity. The relevant definitions should be written out and the difference between continuous and uniformly continuous should be understood. The expression "delta=epsilon/k" contains all the necessary information for the proof.
barksdalemc

## Homework Statement

Show that Lipschitz continutity imples uniform continuity. In particular show that functions sinx and cosx are uniformly continuous in R.

## The Attempt at a Solution

I said that if delta=epsilon/k that Lipschitz continuity imples continuity. Now I am stuck as to how to show uniform continuous.

What are the relevant definitions? Note that Lipschitz continuity is a global property of a function, and uniform continuity is the global version of the local property of ordinary continuity, so it should be straightforward to modify your proof for this case (if it doesn't already apply).

As StatusX said, write out the definitions. What is the difference between "continuous" and "uniformly continuous"? (Your "delta=epsilon/k" contains everything you need.)

## 1. What is Lipschitz continuity?

Lipschitz continuity is a mathematical concept used to describe the smoothness of a function. A function is considered Lipschitz continuous if there exists a constant k such that the distance between the values of the function at any two points is always less than or equal to k times the distance between the points. In simpler terms, this means that the function does not have any sudden, sharp changes in its values and is instead relatively smooth.

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

While both Lipschitz continuity and uniform continuity describe the smoothness of a function, they differ in their definitions and applications. Lipschitz continuity requires the existence of a specific constant k, whereas uniform continuity only requires that for any given distance, there exists a corresponding distance such that the values of the function at those points are within that distance. Additionally, Lipschitz continuity is a stronger condition than uniform continuity, as any Lipschitz continuous function is also uniformly continuous, but the reverse is not always true.

## 3. How is the function sinx Lipschitz continuous?

The function sinx is Lipschitz continuous on the interval [0,2π] because it has a bounded derivative. This means that there exists a constant k such that the absolute value of the derivative of sinx is always less than or equal to k. In this case, the constant k would be equal to 1, as the derivative of sinx is cosx, which has a maximum absolute value of 1 on the interval [0,2π].

## 4. Can you provide an example of a function that is uniformly continuous but not Lipschitz continuous?

Yes, the function f(x) = √x is uniformly continuous on the interval [0,1], but it is not Lipschitz continuous. This is because while the values of the function do not have any sudden jumps or changes, the derivative of the function is unbounded, meaning that there is no constant k that can satisfy the definition of Lipschitz continuity.

## 5. How can the properties of sinx and cosx be shown in R using Lipschitz continuity and uniform continuity?

Both sinx and cosx are Lipschitz continuous on any finite interval, meaning that they are relatively smooth functions with no sudden changes in their values. Additionally, they are also uniformly continuous since for any given distance, there exists a corresponding distance such that the values of the functions at those points are within that distance. This can be shown in R by graphing the functions and observing their behavior, as well as by verifying the definitions of Lipschitz continuity and uniform continuity for these functions using their derivatives.

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