Proving Uniform Continuity using ε-δ

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Today we are going to proof uniform continuity of the funktion cos(x)+x. For this we make use of the abstract epsilon delta definition and use some spicy integral approximations to deal with everything. Enjoy! =D

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When I saw ε-δ in the thumbnail, the distance between my finger and the screen approached zero from above

ishaanvatus

4:32 "It just makes sense" If Fermat had just written that in the margin, Andrew Wiles would have saved himself a ton of work

neilgerace

"you have to play around with the expressions like you would play around with your girlfriend at home" you earned yourself a subscribe

leftistadvocate

Ah, a good ol epsilon delta proof. Just like the neanderthals had to do it. Luckily the ancient egyptians figured out that these proofs can be simplified for differientiable functions if the derivative is bounded.

gdsfish

5:08
Papa Flammy: "Your Girlfirend at Home"
Me: "What Girlfriend?"

janus

Came only to hear "I just kicked a child", job done.

bilyanaconsulova

Epsilon-delta proofs always confused the hell out of me, but you explained that you need to cast some spells to get delta in terms of epsilon and then you're all good. Better than any of my lecturers.

HolyMith

Dang, now not that sounds like really strong analysis and intricate proof right here. Appreciate the information and enlightening indeed!

RCSmiths

You could've posted this before my exams on this shit, not two days after

northernberger

Nice proof💚✨
U can also use The mean value theorem in order to evaluate |cos x -cos y|
Simply, the theorem says that there exists c in [y, x] such that
f'(c)(x-y)=f(x)-f(y)
If we consider our f is cosx, so we get
-sin(c)(x-y)=cos x- cos y
Hence |cos x-cos y|=|sin(c)(x-y)|≤|x-y| since |sin(c)|≤1

djalalmaster

It's so sexy to hear a competent mathematician speak/write in complete mathematical sentences.

k.c.sunshine

0:09
Mathematicians kick children just for fun. xD

diaz

When I saw the sigma and delta I thought this was a chemistry video from another channel about sigma bonds. btw papa can you do a video about TOPOLOGY next

Edit: realized this is an eplison 🤦‍♂️

thedarksword

I had to double check to make sure I clicked on the right video. This was brilliant

HaiNguyen-czbj

I hate maths but I watch your videos anyway. I love watching things that I dont understand at all

djordjesankovic

I think it is a kind of circular reasoning when you use integral to prove continuity.

yuanmingluo

There's no doubt in my mind that you would be able to explain this much better than my analysis professor in your sleep.

Skylitzz_

i love you papa, for the next assignment we have to show that every continuous func. is also e-d-continous. so this for sure does help a lot!

walterodimm

Alternatively, f'(x)=1-sinx => 0<=1-sinx<=2 => -2<=1-sinx<=2 <=> |f'(x)|<=2. So f is Lipschitz which implies uniform continuity.

epeseferma

By the Mean Value Theorem, there exists a point c∈(y, x) such that
|(cosy - cosx)/(y - x)| <= |sinc| <= 1.
So, |cosy - cosx| <= |y - x|

davidbrisbane

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