Abs Continuous Function w/ Unbounded Derivative on [a,b]

In summary, the function f: [-1,1] defined by f(x) = x^2sin(1/x^2) for x ≠ 0, f(0) = 0 is continuous and its derivative f'(x) = 2xsin(1/x^2)-2/xcos(1/x^2) for x ≠ 0, f'(0) = 0 is unbounded on [-1,1]. But this function isn't absolutely continuous...
  • #1
35
0
What is an example of an absolutely continuous function on [a,b] whose derivative is unbounded?

I know that the function f: [-1,1] defined by f(x) = x^2sin(1/x^2) for x ≠ 0, f(0) = 0 is continuous and its derivative f'(x) = 2xsin(1/x^2)-2/xcos(1/x^2) for x ≠ 0, f'(0) = 0 is unbounded on [-1,1]. But this function isn't absolutely continuous...

Any help would be much appreciated : )
 
Physics news on Phys.org
  • #2


We can construct another example along the same lines. Consider the following:

[tex]f(x)=\begin{cases} x^2 \sin (|x|^{-3/2}) & \text{if }x \neq 0 \\ 0 & \text{if } x=0 \end{cases}[/tex]

Computing the derivative yields:

[tex]f'(x) = \begin{cases} 2x \sin (|x|^{-3/2}) - \frac{3}{2} \operatorname{sgn}(x) |x|^{-1/2} \cos(|x|^{-3/2}) & \text{if } x \neq 0 \\ 0 & \text{if } x=0 \end{cases}[/tex]

Since the derivative is integrable on [-1, 1], f is absolutely continuous on that interval.
 
  • #3


Thank you for your help. One question: How I do know that the derivative of that function is integrable? If f' were bounded, then the fact that it is only discontinuous at x = 0 would make f' Reimann integrable, and Reimann integrability implies Lebesgue integrability for bounded functions. But in this case, f' is not bounded, so how do I show that it is Lebesgue integrable?
 
  • #4


f' is obviously measurable, so you only have to show that the integral of the absolute value is finite. But we have:

[tex]\begin{align}\int_{-1}^{1} |f'(x)| \ dx & \leq \int_{-1}^{1} 2|x| |\sin(|x|^{-3/2})| + \frac{3}{2}|x|^{-1/2} |\cos(|x|^{-3/2})| \ dx \\ &\leq \int_{-1}^{1} 2 + \frac{3}{2} |x|^{-1/2} \ dx \\ & = 4 + \frac{3}{2} \int_{-1}^{1} |x|^{-1/2}\ dx \\ & = 4 + 3 \int_{0}^{1} x^{-1/2} dx \\ & = 4 + 6 \sqrt{x} \Big\vert_{0}^{1}\\ & = 10 < \infty \end{align}[/tex]
 
  • #5


Aha. Thank you! I knew that if a bounded function is Reimann integrable then the Reimann integral and the Lebesgue integral of the function are equal. However, I always forget that this can be extended to unbounded functions as long as the Reimann integral is finite.

Thanks again!
 
1)

What is an absolute continuous function with an unbounded derivative on [a,b]?

An absolute continuous function with an unbounded derivative on [a,b] is a function that is continuous on the interval [a,b] and has a derivative that is unbounded (approaches infinity) at every point within that interval. In other words, the rate of change of the function is constantly increasing or decreasing at an unbounded rate.

2)

How is an absolute continuous function with an unbounded derivative different from a regular continuous function?

A regular continuous function has a bounded derivative, meaning that the rate of change of the function is limited or finite at every point within the interval. An absolute continuous function with an unbounded derivative, on the other hand, has a derivative that is not limited and can approach infinity at any point within the interval.

3)

What is the significance of an absolute continuous function with an unbounded derivative?

An absolute continuous function with an unbounded derivative has important applications in mathematics, particularly in the study of integration and measure theory. It can also be used to model real-world phenomena where the rate of change is constantly increasing or decreasing at an unbounded rate.

4)

Can an absolute continuous function with an unbounded derivative exist on a closed interval [a,b]?

Yes, an absolute continuous function with an unbounded derivative can exist on a closed interval [a,b]. As long as the function is continuous on the interval and has an unbounded derivative at every point within that interval, it meets the criteria for an absolute continuous function with an unbounded derivative on [a,b].

5)

How can an absolute continuous function with an unbounded derivative be graphically represented?

An absolute continuous function with an unbounded derivative can be graphically represented as a curve that is continuously increasing or decreasing at an unbounded rate. This can be visualized as a curve that gets steeper and steeper as it approaches certain points on the graph, indicating that the derivative is approaching infinity at those points.

Suggested for: Abs Continuous Function w/ Unbounded Derivative on [a,b]

Replies
4
Views
161
Replies
5
Views
895
Replies
26
Views
1K
Replies
14
Views
1K
Replies
1
Views
746
Replies
25
Views
2K
Replies
10
Views
543
Back
Top