Is a uniform limit of absolutely continuous functions absolutely continuous?

In summary, the conversation discussed the claim that a uniform limit of absolutely continuous functions is absolutely continuous. The sequence of functions that converges to the Cantor function on [0,1] was brought up, and it was confirmed that each of those functions is absolutely continuous and converges uniformly to the Cantor function. However, it was pointed out that the Cantor function itself is not absolutely continuous. The conversation also mentioned the possibility of the Cantor function being a uniform limit of absolutely continuous functions, and it was concluded that this is indeed the case.
  • #1
AxiomOfChoice
533
1
I was reading a Ph.D. thesis this morning and came across the claim that "a uniform limit of absolutely continuous functions is absolutely continuous." Is this true? What about the sequence of functions that converges to the Cantor function on [0,1]? Each of those functions is absolutely continuous, right? And they converge uniformly to the Cantor function, right? But the Cantor function is the canonical example of a continuous, increasing function that's not absolutely continuous!

Just so you know what I mean, I'm talking about the sequence [itex]\{ g_n(x) \}[/itex], where [itex]g_n[/itex] is constant on the middle-third that is removed in stage [itex]n[/itex] of constructing the Cantor set, and linear everywhere else. For example:

[tex]
g_1(x) = \begin{cases} \frac{3x}{2}, & x\in [0,1/3],\\ \frac 12, & x\in (1/3,2/3),\\ \frac{3x}{2} - \frac 12, & x\in [2/3,1]. \end{cases}
[/tex]

This guy is clearly the integral of his derivative, so I think it's reasonable to conclude that each of the other ones is, too.
 
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  • #2
(As an aside...isn't it true that any function that is some piecewise combination of constant and linear functions is absolutely continuous on a closed, bounded interval?)
 
  • #3
I haven't checked it out, but is the Cantor function a UNIFORM limit?
 
  • #5
AxiomOfChoice said:
(As an aside...isn't it true that any function that is some piecewise combination of constant and linear functions is absolutely continuous on a closed, bounded interval?)

Yes, that is correct (assuming that the segments meet at the endpoints, of course). And your reasoning in the OP is correct as well. Uniform convergence of absolutely continuous functions does not, by itself, imply convergence. So unless your Ph. D. included some other condition on the functions (such as the derivatives also converging uniformly, or at least being uniformly bounded above by some integrable function), then (s)he is quite in error. Hopefully their thesis did not depend on that assertion!
 
  • #6
TylerH said:
This Cantor function isn't absolutely continuous. http://en.wikipedia.org/wiki/Cantor_function
You misunderstood my comment. I was wondering iif the Cantor function was a UNIFORM limit of absolutely continuous functions. I am well aware of the fact that it is not absolutely continuous.
 
  • #7
I checked further. The Cantor function is the uniform limit of absolutely continuous functions.
 

FAQ: Is a uniform limit of absolutely continuous functions absolutely continuous?

What is a uniform limit of absolutely continuous functions?

A uniform limit of absolutely continuous functions is a function that is the limit of a sequence of absolutely continuous functions in a uniform manner. This means that for any given epsilon, the difference between the limit function and each of the functions in the sequence is less than epsilon for all points in the domain.

What does it mean for a function to be absolutely continuous?

A function is absolutely continuous if it satisfies the following condition: for any given epsilon, there exists a delta such that for any finite collection of disjoint intervals in the domain whose total length is less than delta, the sum of the absolute values of the differences between the function values at the endpoints of the intervals is less than epsilon.

Why is it important to know if a uniform limit of absolutely continuous functions is also absolutely continuous?

Knowing if a uniform limit of absolutely continuous functions is also absolutely continuous is important because it allows us to make conclusions about the behavior of a sequence of functions based on the behavior of its limit function. This can be useful in various areas of mathematics, such as in the study of differential equations or in the development of numerical methods.

Is every uniform limit of absolutely continuous functions absolutely continuous?

No, not every uniform limit of absolutely continuous functions is absolutely continuous. There are certain conditions that must be met in order for a uniform limit of absolutely continuous functions to be absolutely continuous, such as the domains of the functions in the sequence being bounded or the limit function being continuous.

How can we prove that a uniform limit of absolutely continuous functions is absolutely continuous?

To prove that a uniform limit of absolutely continuous functions is absolutely continuous, we can use the definition of absolute continuity and the properties of limits to show that the limit function satisfies the condition for absolute continuity. We can also use the fact that absolute continuity is preserved under uniform convergence, meaning that if a sequence of functions is uniformly convergent, then its limit function will also be absolutely continuous.

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