Can you visually understand absolute continuity of a function over an interval?

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In summary, uniform continuity and absolute continuity are two different properties of a function. Uniform continuity is a global property, while absolute continuity is a local property and is equivalent to having a derivative. Absolute continuity guarantees that the function has a derivative, and this can be further explored in advanced calculus texts.
  • #1
cappadonza
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i'm having a difficult time trying to grasp what absolute continuity means, i understand uniform continuity. i can't seem to distinguish between the them.
to me it seems that if [tex] f [/tex] on some inteval [tex] [a,b] [/tex] is uniformly continuous then it would be absolutely continuous ?
is there a visual way of describing/thinking about absolute continuity of a function over some interval
 
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  • #2
Uniform continuity is a global property of a function, that is when using the basic definition, you use the same (δ,ε) for all x in the interval.

Absolute continuity is a local property and is equivalent to having a derivative, which can be integrated to get the original function back.
 
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  • #3
thanks for you reply, so absoulte continuity guarantee's the functions has a derivative then ?
if so why, are there any good resources or books, where this is proved/explained in more detailed
 
  • #4
I don't have any references but any good advanced calculus text will have a full discussion.
 

What is absolute continuity?

Absolute continuity is a mathematical concept that describes the relationship between two sets of data. It means that one set of data can be transformed into another set through a process known as integration.

What is the difference between absolute continuity and uniform continuity?

The main difference between absolute continuity and uniform continuity is that absolute continuity deals with the relationship between two sets of data, while uniform continuity deals with the behavior of a function on a single set of data.

How is absolute continuity related to the concept of differentiability?

Absolute continuity is closely related to the concept of differentiability. A function is absolutely continuous if and only if it is differentiable almost everywhere. This means that the derivative of an absolutely continuous function exists except on a set of measure zero.

What are some real-world applications of absolute continuity?

Absolute continuity has many real-world applications in fields such as physics, engineering, and economics. For example, it is used to study the motion of particles, model fluid flow, and analyze economic data.

How can one prove that a function is absolutely continuous?

To prove that a function is absolutely continuous, one must show that it satisfies the definition of absolute continuity. This involves showing that for any given epsilon, there exists a delta such that the sum of the lengths of the intervals in the function's domain is less than epsilon when the corresponding intervals in the function's range are added.

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