Real Analysis | The uniform continuity of sqrt(x).

We show that the square root function is uniformly continuous on its domain.
Please Subscribe: ua-cam.com/users/michaelpennma...
Merch: teespring.com/stores/michael-...
Personal Website: www.michael-penn.net
Randolph College Math: www.randolphcollege.edu/mathem...
Randolph College Math and Science on Facebook: / randolph.science
Research Gate profile: www.researchgate.net/profile/...
Google Scholar profile: scholar.google.com/citations?...
If you are going to use an ad-blocker, considering using brave and tipping me BAT!
brave.com/sdp793
Buy textbooks here and help me out: amzn.to/31Bj9ye
Buy an amazon gift card and help me out: amzn.to/2PComAf
Books I like:
Sacred Mathematics: Japanese Temple Geometry: amzn.to/2ZIadH9
Electricity and Magnetism for Mathematicians: amzn.to/2H8ePzL
Abstract Algebra:
Judson(online): abstract.ups.edu/
Judson(print): amzn.to/2Xg92wD
Dummit and Foote: amzn.to/2zYOrok
Gallian: amzn.to/2zg4YEo
Artin: amzn.to/2LQ8l7C
Differential Forms:
Bachman: amzn.to/2z9wljH
Number Theory:
Crisman(online): math.gordon.edu/ntic/
Strayer: amzn.to/3bXwLah
Andrews: amzn.to/2zWlOZ0
Analysis:
Abbot: amzn.to/3cwYtuF
How to think about Analysis: amzn.to/2AIhwVm
Calculus:
OpenStax(online): openstax.org/subjects/math
OpenStax Vol 1: amzn.to/2zlreN8
OpenStax Vol 2: amzn.to/2TtwoxH
OpenStax Vol 3: amzn.to/3bPJ3Bn
My Filming Equipment:
Camera: amzn.to/3kx2JzE
Lense: amzn.to/2PFxPXA
Audio Recorder: amzn.to/2XLzkaZ
Microphones: amzn.to/3fJED0T
Lights: amzn.to/2XHxRT0
White Chalk: amzn.to/3ipu3Oh
Color Chalk: amzn.to/2XL6eIJ

[@neur303]

Thanks for doing all kinds of tutorials! Even ones that might be not as popular!
Very appreciated 🙏❤️

[@kinpatu]

Love your videos. I wish the YT Algorithm had introduced you earlier.

[@pierineri]

In fact |√x-√y|≤√|x-y| for all non-negative x and y. So it's uniformly continuous.

[@amadeutoletole6777]

Excellent lesson!

[@SalomonChing]

This is the information I was looking for, I had many doubts, thank you very much my friend.✨👍

[@felixmiguet3568]

Your videos helps me a lot to read U.A by Stephen Abott, thanks you so much

[@jimskea224]

On the open set of ordinal integers between zeroth and second.

[@yazhineet.s3774]

Great problem sir, waiting for more of your physics videos.

[@greyforget6916]

Thank you so so so much!

[@goodplacetostop2973]

14:15

[@bsuperbrain]

Nope, the 'stop' was cut off. :D

[@adned6281]

The first good place to stop: ua-cam.com/video/FhaFewKyp-Q/v-deo.html

[@goodplacetostop2973]

@@adned6281 Nope, you haven't dug deep enough. Your video is from March 2020 but we can find earlier than that. Look this, this is from January 2020 : ua-cam.com/video/5Xk1SSUuOjU/v-deo.html

[@adned6281]

@@goodplacetostop2973 Wow, you do know your good places to stop.
Guess my assumption about good place to stop continuity was not justified.

[@goodplacetostop2973]

Adned Actually I haven’t found the original good to place... yet. There’s A LOT of videos to go through lol

[@juanfa98]

at 6:22 in your scratch work, aren't you proving that sqrt(x) is a Lipschitz function?
and all Lipschitz functions are uniformly continuos

[@1tryhxrd.567]

But sir what if we choose x in 0,1 and y in 1,+infinity how would your result hold in this case

[@charlottedarroch]

You could have also picked δ = min{δ_1,δ_2,1} and that still confines x and y to one of [0,2] or [1,inf).

[@watchaccount]

what if epsilon = 1000000 and delta1 = delta2 = 10?

[@jonathanjacobson7012]

Daniel, what would you do with x=1.5

[@charlottedarroch]

@@jonathanjacobson7012 If x = 1.5 and |x-y| < δ = min{δ_1,δ_2,1}, then |x-y| < 1, so y is in (0.5,2.5). If y is in (0.5,2], then both x and y are in [0,2]. If instead y is in (2,2.5), then both x and y are in [1,inf). Either way, choosing δ = min{δ_1,δ_2,1} guarantees that for all positive real numbers x,y with |x-y| < δ, either both x,y are in [0,2], or both x,y in [1,inf), which is sufficient for Michael's argument.

[@uniqueideas3314]

Is step function is uniformly continuous

[@josephhajj1570]

What about f(x)=1/x

[@rogerlie4176]

Set y = x + ẟ/2 and see what happens with |f(x) - f(y)| when x -> 0⁺.

[@leonardoavila899]

Creo que Mr.Penn debe ser más ordenado y escribir un poco más grandes las letras.

[@zeravam]

Michael escribe muy bien y claro, pon la pantalla completa para ver las letras más grandes

[@chessematics]

Sir I'm an 8th grader and having problems while factoring cubics and quartics, could you kindly help me

[@ancientwisdom7993]

When a prof is teaching university curriculum, it is not a proper place to ask for help for middle school stuff. There are literally hundreds of middle school math channels out there.
If u want to badly ask questions to Michael, pose specific hard factorization problems on his vids where he posts competition math - have a basic sense of propriety.

[@chessematics]

@@ancientwisdom7993 i am glad to see that you have the BASIC SENSE OF PROPRIETY, but if sir agrees then I don't think that your BASIC SENSE OF PROPRIETY will be of great worth

[@get2113]

Consider going directly to absolute continuity, so a function can be recovered by integrating its derivative. Key for calculus.

[@pierineri]

Very bad

[@backyard282]

All of this complicated equations and dividing into those two cases is unnecessary.
Just use delta = epsilon^2 and you're basically immediately done.

[@JB-ym4up]

UA-cam unsubscribed me. Go figure.