What do I need to know to understand Uniform convergence?

• I
Hi, I started to study the function of Weierstrass (https://en.wikipedia.org/wiki/Weierstrass_function)
And in one part says that the sum of continuous functions is a continuous function.
i understand this but the Limiting case is a different history depend of the convergence, so what i need to know to self-study properly the uniform convergence, pointwise convergence, etc?

chiro
Hey MAGNIBORO.

Get an introductory book on topology that is self-contained [they tell you this sort of thing in the preface and/or initial chapters].

Real analysis requires a proper undergraduate sequence to have the intuition before you go into all the formalities.

MAGNIBORO
Hey MAGNIBORO.

Get an introductory book on topology that is self-contained [they tell you this sort of thing in the preface and/or initial chapters].

Real analysis requires a proper undergraduate sequence to have the intuition before you go into all the formalities.
ok thanks , I will have to continue studying the theme

FactChecker
Gold Member
You need to study a reference to be precise, but here is a rough description:

Absolute[CORRECTION: Uniform]: For any difference you specify, no matter how small (say ε=0.001), there is an integer N (say 1000), where for all i≥N, |fi(x) - f(x)| < ε for all x
Pointwise: Given an x, for any difference you specify, no matter how small (say ε=0.001), there is an integer N (say 1000), where for all i≥N, |fi(x) - f(x)| < ε for that particular x

You can see that pointwise convergence is much weaker than absolute[correction: uniform]. In fact, if the series is only pointwise convergent, where may be no time where the series[correction: sequence] gets close to the limit at every value of x. The series fi(x) = x1/n is only pointwise convergent to f(x) ≡ 1.0 on R-{0}. You can always find x large enough to get keep fi far from 1.0.

Last edited:
S.G. Janssens
You need to study a reference to be precise, but here is a rough description:

Absolute: For any difference you specify, no matter how small (say ε=0.001), there is an integer N (say 1000), where for all i≥N, |fi(x) - f(x)| < ε for all x
Pointwise: Given an x, for any difference you specify, no matter how small (say ε=0.001), there is an integer N (say 1000), where for all i≥N, |fi(x) - f(x)| < ε for that particular x

You can see that pointwise convergence is much weaker than absolute. In fact, if the series is only pointwise convergent, where may be no time where the series gets close to the limit at every value of x. The series fi(x) = x1/n is only pointwise convergent to f(x) ≡ 1.0 on R. You can always find x large enough to get keep fi far from 1.0.
While the essence of what you say is correct, I have three remarks.

1. Absolute convergence is not the same as uniform convergence. (The former talks about series. The latter talks about sequences or series, where in the second case the series is interpreted as a sequence of partial sums.)

2. The ##f_i## in your example constitute a sequence on ##\mathbb{R}##, not a series.

3. This sequence does not converge to ##1## at ##x = 0##.

Last edited:
FactChecker
FactChecker
Gold Member
While the essence of what you say is correct, I have three remarks.

1. Absolute convergence is not the same as uniform convergence. (The former talks about series. The latter talks about sequences or series, where in the second case the series is interpreted as a sequence of partial sums.)

2. The ##f_i## in your example constitute a sequence on ##\mathbb{R}##, not a series.

3. This sequence does not converge to ##1## at ##x = 0##.
I stand corrected. Thanks. I got sloppy. I should not try to answer these while watching a football game.

S.G. Janssens
Thank you all for your comments, For these I understood the epsilon definition like make the difference or error between |fi(x) - f(x)| Arbitrarily small
I need to study more the epsilon definition and the sup and inf limits.
Many times i saw them And I guess As it seems is the key to many things in math

S.G. Janssens