Converg. Seq. of Functions, Derivatives Bounded, Limit not Differentiable

In summary, the task is to find a sequence of differentiable functions that satisfy certain conditions and have a convergent subsequence that is not differentiable. This can be proven using Arzela-Ascoli for the reals. A possible example is the series of functions f(x)=|x| which is not differentiable, and can be approximated by the differentiable function sqrt(x^2).
  • #1

Homework Statement


Find a sequence of differentiable functions $f_n\colon [a,b]\rightarrow\mathbb(R)$ s.t.:
--there exists $M>0$ with $|f_n'(x)|\leq M$ for all $n\in\mathbb{N}$ and $x\in[a,b]$;
--for all $n\in\mathbb{N}$, $|f_n(a)|\leq M$;
--$(g_n)$ is a convergent subsequence with $lim_{n\rightarrow\ifty}g_n(x)=g$ for $f$ NOT DIFFERENTIABLE.

Homework Equations


Arzela-Ascoli for the reals.

The Attempt at a Solution



I have already proved, using Arzela-Ascoli, that such a $g$ exists for any sequence $(f_n)$ fulfilling the first two conditions. But I simply cannot come up with a concrete example where the limit is not differentiable!

Thanks!
 
Physics news on Phys.org
  • #2
f(x)=|x| is not differentiable. Can you think of a series of differentiable functions that converge to it? Hint, |x|=sqrt(x^2).
 

What is a convergent sequence of functions?

A convergent sequence of functions is a sequence of functions that approaches a single limit function as the independent variable approaches a certain value. This means that as the input value gets closer and closer to a specific value, the output of the sequence of functions also gets closer and closer to a specific function.

What does it mean for derivatives to be bounded?

If a derivative is bounded, it means that its values are limited and cannot exceed a certain value. In other words, the derivative does not grow infinitely large or small, but instead stays within a certain range of values.

Can a limit not be differentiable?

Yes, a limit can be not differentiable. This occurs when the derivative of a function does not exist at a specific point because the function is not continuous or has a sharp turn or corner at that point.

What is the relationship between a convergent sequence of functions and bounded derivatives?

A convergent sequence of functions can have bounded derivatives, meaning that the derivatives of each function in the sequence are limited and do not grow infinitely large or small. However, it is also possible for a convergent sequence of functions to have unbounded derivatives, meaning that the derivatives of each function in the sequence do not have a limit and can grow infinitely large or small.

How does the concept of a convergent sequence of functions and bounded derivatives relate to real-world applications?

In real-world applications, a convergent sequence of functions with bounded derivatives can represent a system or process that is approaching a steady state or equilibrium. This can be seen in fields such as physics and economics, where functions and their derivatives are used to model and predict the behavior of physical systems and economic trends.

Suggested for: Converg. Seq. of Functions, Derivatives Bounded, Limit not Differentiable

Replies
16
Views
1K
Replies
5
Views
1K
Replies
7
Views
1K
Replies
5
Views
663
Replies
8
Views
532
Replies
7
Views
554
Back
Top