Real Analysis 24 | Pointwise Convergence

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This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.

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00:00 Intro
00:33 Pointwise convergence
02:03 1st Example
03:20 2nd Example
06:40 3rd Example
07:43 Credits

#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[raoufmedj]

I've been searching YouTube for a detailed well-presented explanation of this notion, and I must say, this is literally the best one out there. Great content!

[subhradeepghosh]

I was having some problem understanding pointwise convergence. This video cleared all my doubts. This is the best maths channel that I've found on YouTube till date.

[malikialgeriankabyleswag]

You must have had great professors for your undergrad because the little detail about fixing the x's such that f_n(x)'s become a sequence in the real numbers as opposed to leaving x arbitrary so that f(x)'s are a sequence in the set of functions, this helped me to understand the real essence of the two kinds of convergence, better than anything else I've heard.. Very helpful! Thank you and God bless.

[CrimsonSquaredX]

Thatnk you for the illuminating examples and illustrations. I now understand that I had deeply undervalued the need for the notion of uniform convergence.

[angelmendez-rivera]

An interesting and simple example of pointwise convergence where the convergence appears weak is with the sequence f[n] of functions (–1, 1] —> R, (f[n])(x) = x^n. f : (–1, 1] —> R defined by f(x) := lim f[n] (n —> ♾) satisfies f(x) = 0 for every x in (–1, 1), but f(1) = 1. This is my favorite example, because it is quite an intuitive sequence to consider, and one which you expect to have nicer convergence properties.

[travist]

Finally an explanation with visuals that makes sense!

[x.s]

Hi Thanks a lot for this wonderful video. For example 2, i did get where is the case when x>0, fn(x) is not 0 for n <1/x

[berke-ozgen]

Thank you for the content. I thought that 2nd example is not a good selection for this topic. Let me explain: when x goes to 0, fn(x) goes to infinity for increasing values of n. I am not sure that this function is even convergent. I guess it has 2 sub function sequences which convergences 0 and 1, respectively, which makes it non convergent.

[Hold_it]

Great video! Can't wait for the one on uniform convergence :D

[chendian-jing]

For the second example "n^2x(1-nx)", I am a bit not sure. By definition, it requires for all epsilon > 0. Obviously, we cannot choose epsilon < 1/4 since in this case n = 1. If you choose epsilon = 1/8 for example, it implies n = 1/2. In this case 1/n = 2 > 1 which exceeds the domain [0, 1]. Is it still pointwise convergent? I am confused.

[zyzhang]

I don’t understand why for the second example for x=0 the value is not undefined. Anyone can explain?

[predatoryanimal]

i have taken the integral of the function in the example (n^2 x (1- nx)) as two integrals n^2 x dx - n^3 x^2 dx from 0 to 1/n and it gave me a value of 1/6, does that mean that at the limit n-> infinity, there is an f(x) in the 0+ vicinity whose value is 1/6, or i have done something wrong? sorry for constantly bothering you with these questions but math is really tricky for me, thx for the answer in advance!

[pirzadaaakib]

Sir, could you please tell me the best book for convergence topic

[hustle]

You have to relate this topic to anything that makes this topic easier, I can't understand,

[hustle]

But you should do more efforts to make topic very easy