(Global) Modulus of Continuity

Authors: Brittney D’Oleo, Elise Favia

Date: October 27, 2017

 

Introduction:

This week’s topic, (global) modulus of continuity is an extension of last week’s topic, the (local) modulus of continuity. In the blog, we will explore the new definition and compare it to last weeks definitions. We will also prove an interesting result creating an equivalence between uniform continuity and having a (global) modulus of continuity, as well as some applications involving power functions.

 

Review:

As a quick summary, last week’s blog provided two definitions (this is Rhiannon and Shuyi’s precise wording):

  1.  Definition: A function \sigma :[0,\infty)\rightarrow [0,\infty) is a modulus of continuity if \sigma(0)=0 and \lim_{d\rightarrow 0^{+}}\sigma(d)=0.
  2.  Definition: f: A \rightarrow \mathbb{R} is continuous at c \in A with (local) modulus of continuity \sigma if there exists \delta >0 so that |f(x)-f(c)| \leq \sigma(|x-c|) for |x-c| < \delta.

Essentially, the modulus of continuity is a way of measuring the rate of convergence, similar to, but not the same as the slope or derivative of the function. A result from last week’s blog was that \sigma_1=md is a local modulus of continuity for f at c if m\geq f'(c).

 

New Definitions:

There is a difference between the definition of local modulus of continuity and global modulus of continuity, but before I address this, I will formally write the definition for you.

Definition: A function f: A \rightarrow \mathbb{R} has a global modulus of continuity \sigma, if \exists \delta>0 so that |f(x)-f(y)| \leq \sigma (|x-y|) for all x,y \in A such that |x-y|<\delta.

From the review section, when determining if a function f is defined to have local modulus of continuity \sigma, there is a focus at a particular point c\in A. In contrast, when determining if a function f is defined to have global modulus of continuity \sigma if we can find one \delta where |f(x)-f(y)| \leq \sigma (|x-y|) for any two points x,y \in A where the distance between those two points are within a \delta window of each other.

 

Looking at Uniform Continuity Through a Different Lens:

The definition of uniform continuity states that a function f: A \rightarrow \mathbb{R}
is uniformly continuous on its domain A if for every \epsilon>0, there exists a \delta>0 such that for all x,y \in A, |x-y| < \delta implies that |f(x)-f(y)|< \epsilon. This definition is very similar to that of global modulus continuity, the main difference is that global modulus continuity has this variable \sigma. In fact, \sigma has some explicit functional dependence on \epsilon and \delta. If we look at \sigma(\delta) : = \sup\{|f(x)-f(y)|: |x-y|<\delta\} then these definitions are actually equivalent. To motivate this lets look at the same three functions we did in class, but instead of fixing \epsilon, lets fix \delta =1.

In this function notice that no matter what size \delta window we pick, the \epsilon window will be constant, and so we can find a supremum \epsilon window so that for any two points in the domain such that the the distance between the points is less than \delta, f(x) and f(y) are in this “largest” \epsilon window.

 

In this function notice that that if we shift our \delta window the right, the \epsilon window will constantly increase, and so we cannot find a supremum \epsilon window for any two points in the domain such that the the distance between the points is less than \delta.

 

In this function notice that no matter what \delta window we choose, the \epsilon window will constantly shrink towards 0 as you increases x, and so we can find a supremum \epsilon window for any two points in the domain such that the the distance between the points is less than \delta. This will simply be the \epsilon window around the point 0 corresponding to the given \delta window.

Notice that figures 1 and 3 have global modulus continuity \sigma(\delta) : = \sup\{|f(x)-f(y)|: |x-y|<\delta\} and that these two functions are also uniformly continuous. Instead of looking at a set \epsilon window and determining if we can find an infimum \delta window that satisfies the definition of uniform continuity, we are looking at a set \delta window and seeing if we can find a supremum \epsilon window that satisfies the definition of the global modulus of continuity definition. These are equivalent statements and so the take away of this section should be that if a function f:A \rightarrow \mathbb{R} has a global modulus of continuity, than f is also uniformly continuous and vice versa.

 

Extension of Power Function:

There are several families of (global) moduli functions that are considered Power Functions. In this section, k>0,p>0

A power function, \sigma(d)=kd^p is a Hölder condition if it is (global) modulus of continuity. Such functions satisfy the condition |f(x)-f(y)|\leq k|x-y|^p for all x,y in the domain of f

A function with modulus \sigma(d)=kd for all x,y in the domain is called a Lipshitz function.
Notice that this is a special case of the Hölder condition with p=1. We have |f(x)-f(y)|\leq k|x-y|^p from the Hölder condition, and so |f(x)-f(y)|\leq k|x-y|^1=k|x-y|.
All Lipshitz functions therefore satisfy the Hölder condition.
Another important observation is that |f(x)-f(y)|\leq=k|x-y|\rightarrow\frac{|f(x)-f(y)|}{|x-y|} \leq k. In other words, the derivative (or slope) of the function is bounded by k.

 

Works Cited:

https://www.revolvy.com/main/index.php?s=Modulus%20of%20continuity&item_type=topic

https://www.revolvy.com/main/index.php?s=Hölder%20condition&uid=1575

Uniform continuity done right

Posted on by Brittney D'Oleo
Categories: Weekly Blogs

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