# Punctual and uniform convergence

• Kernul
In summary: Why if setting ##\epsilon = 0.05##, we have ##f_n(x) = 0.9##?The function ##f_n## goes through all values in the interval ##]0,1]## no matter what the number ##n## is, because it's a continuous function. Therefore we know that for any ##n##, there are points where the function's value differs from the limit ##g(x)## by more than the number ##\epsilon = 0.05## (note that the function ##g## always has value 1, 1/e or 0). This is just a single example that is sufficient to prove that the function sequence ##f_n## is not uniformly converging
Kernul

## Homework Statement

##I## a set of real number and ##f_k : I \rightarrow \mathbb{R}## a succession of real functions defined in ##I##.
We say that ##f_k## converges punctually in ##I## to the function ##f : I \rightarrow \mathbb{R}## if
$$\lim_{k \to \infty} f_k(x) = f(x), \hspace{1cm} \forall x \in I$$
Which is like saying:
$$\forall \epsilon > 0,\hspace{1mm} \forall x \in I,\hspace{1mm} \exists \nu_{\epsilon, x} \in \mathbb{R} : |f_k(x) - f(x)| < \epsilon \hspace{1cm} \forall k > \nu_{\epsilon, x}$$

## The Attempt at a Solution

What is the ##\nu_{\epsilon, x}##? Is it the same in the normal succession of real numbers? Could someone make some examples?
And I've read that for the uniform convergence, this ##\nu_{\epsilon, x}## depends only from ##\epsilon##, so ##\nu_{\epsilon}##, and that the uniform convergence implies the punctual convergence but not vice versa. Any example with these too?

Kernul said:

## Homework Statement

##I## a set of real number and ##f_k : I \rightarrow \mathbb{R}## a succession of real functions defined in ##I##.
We say that ##f_k## converges punctually in ##I## to the function ##f : I \rightarrow \mathbb{R}## if
$$\lim_{k \to \infty} f_k(x) = f(x), \hspace{1cm} \forall x \in I$$
Which is like saying:
$$\forall \epsilon > 0,\hspace{1mm} \forall x \in I,\hspace{1mm} \exists \nu_{\epsilon, x} \in \mathbb{R} : |f_k(x) - f(x)| < \epsilon \hspace{1cm} \forall k > \nu_{\epsilon, x}$$

## The Attempt at a Solution

What is the ##\nu_{\epsilon, x}##? Is it the same in the normal succession of real numbers? Could someone make some examples?
And I've read that for the uniform convergence, this ##\nu_{\epsilon, x}## depends only from ##\epsilon##, so ##\nu_{\epsilon}##, and that the uniform convergence implies the punctual convergence but not vice versa. Any example with these too?
You may consider the sequence ##f_k := \max \{k-k^2\cdot \,\vert \,x-\frac{1}{k}\,\vert \, , 0\}## and ##f(x)=0## as its limit.

Covergence here is ##f_k \longrightarrow f##. That is ##f_k(x) \longrightarrow f(x)##. Now the ##\nu## define in a way how fast this is because they (usually) become bigger the smaller the ##\epsilon## is. So the difference between punctual and uniform convergence is basically whether this "speed of convergence" is the same at all points ##x##, in which case we call it uniform, or whether this "speed of convergence" depends on where we are on the ##x-##axis, in which case it is only punctual.

Kernul
One example of punctual converge is how the function ##f_{n}(x) = e^{-x^{2n}}## approaches the function "##g : g(x) = 1## if ##|x|<1##, ##g(x) = 1/e## if ##|x| = 1##, ##g(x) = 0## if ##|x|>1##" when the natural number ##n## grows without bound. If you set ##\epsilon = 0.05##, there will always be a point ##x## for which ##f_{n}(x) = 0.9## and outside the chosen bounds, no matter how large ##n## is.

An example of uniform convergence is how the sequence ##f_{n}(x) = x + 1/n## approaches ##g(x) = x## when ##n\rightarrow \infty##

Last edited:
Kernul
Kernul said:
punctual convergence
I've never seen it described as "punctual" convergence. The English words point and punctual are derived from Latin "punctum," but the usual meaning of punctual is "on time," as in a punctual arrival of a student to a class.

The type of convergence discussed in this thread is called "pointwise" convergence in English.

Kernul
fresh_42 said:
You may consider the sequence ##f_k := \max \{k-k^2\cdot \,\vert \,x-\frac{1}{k}\,\vert \, , 0\}## and ##f(x)=0## as its limit.

Covergence here is ##f_k \longrightarrow f##. That is ##f_k(x) \longrightarrow f(x)##. Now the ##\nu## define in a way how fast this is because they (usually) become bigger the smaller the ##\epsilon## is. So the difference between punctual and uniform convergence is basically whether this "speed of convergence" is the same at all points ##x##, in which case we call it uniform, or whether this "speed of convergence" depends on where we are on the ##x-##axis, in which case it is only punctual.
Oh, I get the ##\nu## part, but why the limit is ##f(x) = 0##?

hilbert2 said:
One example of punctual converge is how the function ##f_{n}(x) = e^{-x^{2n}}## approaches the function "##g : g(x) = 1## if ##|x|<1##, ##g(x) = 1/e## if ##|x| = 1##, ##g(x) = 0## if ##|x|>1##" when the natural number ##n## grows without bound. If you set ##\epsilon = 0.05##, there will always be a point ##x## for which ##f_{n}(x) = 0.9## and outside the chosen bounds, no matter how large ##n## is.

An example of uniform convergence is how the sequence ##f_{n}(x) = x + 1/n## approaches ##g(x) = x## when ##n\rightarrow \infty##
Why if setting ##\epsilon = 0.05##, we have ##f_n(x) = 0.9##?

Mark44 said:
I've never seen it described as "punctual" convergence. The English words point and punctual are derived from Latin "punctum," but the usual meaning of punctual is "on time," as in a punctual arrival of a student to a class.

The type of convergence discussed in this thread is called "pointwise" convergence in English.
Yes, you're right. I'm sorry but I'm not a native English speaker. When I translated from my native language (Italian) to English, I thought it was correct.

Kernul said:
Oh, I get the ##\nu## part, but why the limit is ##f(x) = 0##?
##f_k(x) = max \{k-k^2\,\vert \,x-\frac{1}{k}\,\vert \,\, , \,0\}##.
So the only values ##f_k(x) \neq 0## are in the interval ##]0,\frac{2}{k}[## which is getting smaller by increasing ##k##. Hence we have only to consider the point ##x_0=0##. But ##\lim_{k \rightarrow \infty}f_k(x_0)=0## because you can always find a zero in a small neighborhood of ##x_0##, although the values at exactly ##x_0## are large.

Kernul
Kernul said:
Why if setting ##\epsilon = 0.05##, we have ##f_n(x) = 0.9##?

The function ##f_n## goes through all values in the interval ##]0,1]## no matter what the number ##n## is, because it's a continuous function. Therefore we know that for any ##n##, there are points where the function's value differs from the limit ##g(x)## by more than the number ##\epsilon = 0.05## (note that the function ##g## always has value 1, 1/e or 0). This is just a single example that is sufficient to prove that the function sequence ##f_n## is not uniformly convergent.

Kernul
The order of quantifiers is important. "...for each ... there exists..." can have a different meaning than "...there exists... for each...".

For example:
"For each instrument X there exists a virtuoso Y such that Y plays X "
versus
"There exists a virtuoso Y such that for each instrument X, Y plays X".

The first statement is a modest claim. The second statement asserts the existence of a fantastic musician who is a virtuoso at playing every instrument.

In pointwise convergence, the definition uses the order "...for each x ...there exists a ##\nu##..."

In uniform convergence, the definition uses the order "...there exists a ##\nu## ...for each x...".

The sequence ##f_k(x)## converges pointwise to ##f(x)## on the interval ##I## is the more modest claim that:
For each ##\epsilon > 0## and for each ##x \in I##, there exists a ##\nu## such that if ##k \gt \nu## then ##|f(x) - f_k(x)| \lt \epsilon##.

The sequence ##f_k(x)## converges uniformly to ##f(x)## on the interval ##I## is the grander claim that:
For each ##\epsilon > 0## there exists a ##\nu## such that for each ##x \in I ##, if ##k \gt \nu## then ##|f(x) - f_k(x)| \lt \epsilon##.

fresh_42 and Kernul

## What is punctual convergence?

Punctual convergence is a type of convergence in which a sequence of functions converges to a specific function at each point in the domain. It is also known as pointwise convergence.

## What is uniform convergence?

Uniform convergence is a type of convergence in which a sequence of functions converges to a specific function in a way that is independent of the specific point in the domain. It is also known as global convergence.

## What is the difference between punctual and uniform convergence?

The main difference between punctual and uniform convergence is the way in which the sequence of functions approaches the limit function. In punctual convergence, the sequence of functions converges to the limit function at each point in the domain. In uniform convergence, the sequence of functions converges to the limit function in a way that is independent of the specific point in the domain.

## Why is uniform convergence important?

Uniform convergence is important because it guarantees that the limit function is continuous. This is not always the case with punctual convergence, where the limit function may not be continuous. Uniform convergence also allows for the interchange of limits and integrals, which is a useful tool in many mathematical and scientific applications.

## How can one determine if a sequence of functions is uniformly convergent?

There are several methods for determining if a sequence of functions is uniformly convergent, including the Cauchy criterion, the Weierstrass M-test, and the Dini's theorem. These methods involve analyzing the behavior of the sequence of functions and comparing it to known convergence criteria.

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