Real Analysis 31 | Uniform Limits of Continuous Functions are Continuous

Please consider to support me if this video was helpful such that I can continue to produce them :)

🙏 Thanks to all supporters! They are mentioned in the credits of the video :)

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.

x

00:00 Intro
00:14 Uniform convergence for sequence of functions
01:09 Theorem for uniform limit of continuous functions
02:18 Proof of the Theorem
07:41 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)

[zoedesvl]

I'd like to add some history.
It was during this time, i.e. when mathematicians studied topics in this video, that they realised the potential to generalise the concept of convergence, like, a point doesn't have to be a real or complex number, it can be a function or something else, and their behaviour is similar to numbers... Maybe we can study convergence regardless of what does a point mean. Modern topology was somewhat inspired by these concepts. It was Poincare who pointed out that topology need to be considered as a branch of mathematics, not some blur words. Decades later the formal definition of topology was given and here we have our modern topology.

[Hold_it]

Thank you for always being so enthusiastic in teaching us mathematics.

[douglasstrother]

My experience with Real Analysis is limited to what is covered in Calculus (single-varible through vector) to validate the basic ideas of continuity, limits, differentiation and integration. As we go through this sequence of videos, I'm convinced that using visual representations of definitions and theorems make Real Analysis much more self-evident.

If these videos only had equations, the concepts would be completely opaque.

[lucaug]

Uniform limits are so nicely behaved, love it! Thanks for another great video!

[douglasstrother]

"ε-tube"!
*I'm* using that, with proper citation, of course.

[maciekkochanowicz]

I think you can draw a graph of f and arbitrary fN, mark four points f(x0), fN(x0), f(x), fN(x) and all three epsilons, that is three sides of a quadrangle generated by this four points, and then visualise the inequality by showing, that the "direct" path from f(x) to f(x0), that is the fourth side of the quadrangle, is shorter than the "detour" via fN(x) and fN(x0), that is the path "via" three other sides.

At least it helped me a lot, when I made such drawing for myself ;-).

[whatitmeans]

Naively speaking every function that represents physical phenomena must be energy limited and time-finite, or in mathematic world, squeared-integrable and compact-supported, Could you make a serie about the properties of this kind of functions?.. I want to know How is limited its maximum possible rate of change (or max slew rate)? It will going to be related to the frequency spectra? like max dg(t)/dt <= int_-inf:inf [ 2*pi*i*f*Fourier{g(t)}*df ] or something similar?

[scollyer.tuition]

Are you planning to cover interchange of limits and integrals at some point? It would seem appropriate now that you've discussed uniform convergence.

[brindhap]

I need example of uniform continuity with graphical