Uniform Continuity (Example 1): The Basics

Uniform Continuity Basic Example

In this video, I work out a basic example of how to show that a function is uniformly continuous

[hypershadic]

Thank you so much for reviewing this subject of analysis, I was really struggling with these definitions!

[wtt]

Sir has given a perfect explanation on uniform continuity in his great video .Thank you sir !

[dylank]

The best thing about uniform continuity is that it's for free for every continuous function defined on a compact space. So, as a little fun fact, f(x) = x^2 is uniformly continuous on every intervall [a, b] (and every other bounded and closed set for that matter)

[arturcostasteiner]

I think it's interesting to notice that every Lipschitz function is uniformly continuous but the converse is not true.

[arturcostasteiner]

On the complex domain, there's an amazing conclusion, which doesn't seem to be very known:

If f is entire and uniformly continuous on C, then f is an affine mapping: for every z, f(z) = az + b, a and b complex constants.

[asamenechbayissa]

I am getting interested in analysis from those videos

[tomasasnes]

I actually understood this!!! Awsome video

[andrewmichel]

I will take real analysis in the near future and am a bit worried about it but these videos make me feel like the class wont completely blindside me

[tomoki-vo]

for my engineering problems i use the perpondicular to the bisector angle between left and right tangents and i trait it as a derivative

[maximelamoureux]

Your video are really helpfull. Can you make a video on several example to prove that a function isn't UC ?

[DynamicMateTV]

Could you please make video explaining uniform continuity visually/diagrmatically?

[theunknownscientist]

Every continuous function on a compact interval is uniformly continous on that interval?! That seems soooo awesome, I can't believe it. Hmm, maybe I'll try a proof? Kinda hard tho.

[him]

You say you’ll cover in an upcoming video how functions on compact sets are uniformly continuous but I first learnt about both those terms from a video you made ages ago! We already have this from you. Just a thought, but the definition of finite covers and compact sets was hard to understand - why is (1, 6] not compact and [1, 6] compact?

[ethancheung]

i always wonder why we dont use the scapework as the proof. because usually in these kind of epsilon-delta proof, everything is bidirectional.
if it is purely about the format of the proof (prefer top to bottom), i think we can do the scapework from bottom to top, voila!

[yoav]

So x^2 is unif cont in any finite interval unless delta=e/inf counts😆

[pierreabbat]

How about f(x)=cube root of x on all R?

[stevenmellemans]

I don’t find this very clear, if my daughter or son would give this as an answer, they would loose a lot of marks on that question. Not by me but but their teacher :-) They should use stricter notation. For instance I didn’t see any <=>.

[colleen]

Use the Chen Lu!




Just kidding I have no idea what this is