How can I find a sequence of functions satisfying certain properties?

In summary, the conversation discusses finding a sequence of differentiable functions $f_n$ that converges uniformly to a differentiable function $f$, but the derivative of $f_n$ only converges pointwise to a function $g$ that is not equal to the derivative of $f$. One suggested method is to consult the book "Counterexamples in Analysis" and use the function $f_n(x)=x/(1+n^2x^2)$ as an example. The speaker mentions that they have seen the solution in the textbook but are unsure of how the sequence of functions was derived.
  • #1
evinda
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Hey! ;) I am looking at the following exercise:
Find a sequence of differentiable functions $f_n$,such that $f_n \to f$ uniformly,where $f$ is differentiable, $f_n' \to g$ pointwise,but $f'\neq g$.

How can I find such a sequence of functions? Is there a methodology to do it?? :confused:
 
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  • #2
One method is to look in the book "Counterexamples in Analysis". Consider $f_n(x)=x/(1+n^2x^2)$. Check the required properties.
 
  • #3
Evgeny.Makarov said:
One method is to look in the book "Counterexamples in Analysis". Consider $f_n(x)=x/(1+n^2x^2)$. Check the required properties.

I saw the solution of the textbook,but I didn't know how they found the sequence of functions $f_n$..
 

FAQ: How can I find a sequence of functions satisfying certain properties?

What is a sequence of functions?

A sequence of functions is a list of functions that are related to each other and share a common domain and range. Each function in the sequence is denoted by a subscript, such as f1, f2, f3, etc. The order of the functions is important and can impact the overall behavior of the sequence.

How do you find a sequence of functions?

To find a sequence of functions, you first need to determine the pattern or relationship between the functions. This can involve looking for common variables, coefficients, or operations. Once the pattern is identified, you can write out the sequence of functions using the pattern.

What are some common applications of sequences of functions?

Sequences of functions have many applications in mathematics and science. They are often used to represent mathematical models of real-world phenomena, such as population growth or the movement of objects. They are also used in calculus to approximate and analyze functions.

How do you prove that a sequence of functions converges?

In order to prove that a sequence of functions converges, you must show that the limit of the functions approaches a specific value as the subscript approaches infinity. This can be done using various methods, such as the squeeze theorem or the Cauchy criterion.

What are some challenges associated with finding a sequence of functions?

One challenge of finding a sequence of functions is determining the correct pattern or relationship between the functions. This can require a lot of trial and error and can be time-consuming. Another challenge is ensuring that the sequence converges and accurately represents the desired function or phenomenon. This may require careful analysis and proof techniques.

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