Real Analysis | Introduction to differentiability.

We introduce the notion of differentiability and the derivative. Next, we give an example and a non-example. Finally, we prove that differentiability at a point implies continuity at that point.
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[@RobsMiscellania]

I am always impressed with how strong of a condition differentiability is. It can act as a sledgehammer in some proofs, but proving differentiability in interesting regions can be delicate. Analysts continually amaze me with their ability to conjure the required inequalities out of thin air. That implies a deep understanding of the properties of numbers.

[@Leomenxd]

I need more videos like this one, simple but yet useful

[@billclinton6040]

This is great stuff. I think it would be a great next lesson to talk about the Mean Value Theorems of differential and integral calculus. I remember these being applied frequently throughout many of my engineering courses.

[@xriccardo1831]

He is going to make them, Just give him 2 seconds

[@tomatrix7525]

Yea, they’re big in most math related courses

[@kavinash7044]

Love your explanations

[@salaprance5283]

I'm taking real analysis right now (using baby Rudin), and seriously I was so lost with the proof on differentiability implying continuity before watching this video. The way you clearly explain each step in the process is profoundly helpful, thank you so much

[@electrovector7212]

This is the video we were looking for

[@borisbryant.007]

in the top ten today thank you mike

[@xXSoutwarDXx]

Great explanation, it just remind me old memories

[@vineetferrari7954]

I'm in year 9 and I don't understand any of this. But I enjoy it

[@azhakabad4229]

Yeah! I was searching for this. Thanks Sir

[@atifatif2616]

Great sir, your lectures always helpful

[@xXSoutwarDXx]

It would be awesome to see you explaning the first theorem of calculus and fractional calculus

[@goodplacetostop2973]

11:44

[@yazhineet.s3774]

You are late 😛 today

[@mjones207]

Kevin Chen's stealing your thunder.

[@yazhineet.s3774]

@@mjones207 yep

[@goodplacetostop2973]

YAZHINEE T.S Michael’s 8PM EST videos are the middle of the night for me. So yeah, some days I’d like to sleep at night 😂

[@Gambogelilly]

Thank you!

[@abdurrahman.alhussni]

Great explanation .. Thank you .. I have a question please .. Let's suppose that the limit ( (f(x)-f(a))/(x-a) ) is infinity when x approaches a from above a and from under a .. then can we claim that f is differentiable on a or not? If not then why (since its not obvious from the defination of the differentiability)?

[@schweinmachtbree1013]

this is called the case of a "vertical tangent". in this case we say that f is not differentiable at a. an example of this is the function f(x) = x^{1/3} (the cube root) at x=0. the reason f is not differentiable at a is because the definition of f being differentiable at a states that the limit of the difference quotient ( (f(x)-f(a))/(x-a) ) exists as x->a *and is finite*. it does feel a little "annoying" that such functions are not differentiable, but think about it this way: if f has a vertical tangent at a then f cannot be well-approximated by a linear function at a (since the line which best approximates f at a is a vertical line, but this does not constitute a function)

[@wise_math]

If it goes to infinity it means it has no limit. Which means no derivative.

[@abdurrahman.alhussni]

@@schweinmachtbree1013 thank you so much for the explanation .. I have another question please .. Let the limit of f when x approaches a from above a is positive infinity and the limit of f when x approaches a from under a is positive infinity also .. Then can we say that f has a limit when x approaches a or not?

[@schweinmachtbree1013]

عبدالرحمن الحصني this is the same as your previous question, but applied to f instead of (f(x)-f(a))/(x-a). In this case we say that the limit of f at a is positive infinity, or that “f diverges to infinity at a” (some would even say “f converges to infinity at a). Note that a lot of the time we need limits to be finite (e.g. with differentiability), and in such a situation we would say “f does not have a finite limit at a” (although it does have a limit at a)

[@abdurrahman.alhussni]

@@schweinmachtbree1013 Thank you so much

[@samli3294]

Even tho i like mathematics..but what really brought me to here is because of that man🤣😍

[@kevinchen8325]

11:44 good place to stop

[@djvalentedochp]

almost

[@JB-ym4up]

A good place

[@mathmeup561]

Thank you for the viseo! Shouldn't we use the definition to prove that derivative of x^n is nx^n rather than calculating using Calculus? This is newton's way of computing derivative! Should not we show |(a+h)^n-a/h-na^{n-1}|

[@algorithmo134]

Make more real analysis video. Cover all rudin chapters please and that would be the best real analysis playlist

[@j.k.priyadharshini9753]

I see you everywhere i go,👀👀👀

[@yazhineet.s3774]

I have a doubt at 5:41 ; why is the absolute value of "a" taken , isn't that point supposed to be the desired value. And whether the left hand limit satisfies the right hand limit too.

[@arvindsrinivasan424]

🔥🔥🔥

[@rounaksinha5309]

Sir, please make a video on location of roots of quadratic equation . I searched the whole KZbin but couldn't find a proof of those result.

[@schweinmachtbree1013]

the proof of the quadratic formula is very well-known. for example see the Quadratic Formula article on wikipedia, specifically the section "Derivations of the formula".

[@rounaksinha5309]

@@schweinmachtbree1013Sir, I am not looking for proof of quadratic formula.I am looking for proof of a set of results mentioned as location of roots in chapter of quadratic equations and expression. Thank you for your help sir. Sorry for bad English. I am from India

[@schweinmachtbree1013]

EduHub India The roots of the quadratic equation ax^2 + bx + c = 0 are located at x = (-b +/- sqrt(b^2 - 4ac))/2a, which is the quadratic formula. I am not sure what you mean by “quadratic expression”. If this is not what you want then I can’t help you, sorry

[@rounaksinha5309]

@@schweinmachtbree1013 thank u for helping me out,sir

[@orenfivel6247]

I wanna show my version of diff⇒cont w\ ε-δ: ∀ε>0, choose δ1>0 s.t |(f(x)-f(a))/(x-a)-f'(a)|< ε ∀x:|x-a|

[@parameshwarhazra2725]

Wtf is it an educational video I just watched an adult and before it started. KZbin this is not a good job. It is totally mind diverting

[@VaradMahashabde]

Ads seems to be largely dependent on the viewer, very weakly on the channel or the video

[@parameshwarhazra2725]

@@VaradMahashabde what do you mean by that buddy???

[@VaradMahashabde]

@@parameshwarhazra2725 KZbin decides which ad to play depending on who is watching, and that is very weakly dependent on what video is being watched

[@parameshwarhazra2725]

@@VaradMahashabde I don't watch this kind of things still why they are showing such things. Disgusting!!!!

[@hybmnzz2658]

xd XD XDDDDDD xdd