Real Analysis | Continuity

We present the precise definition of continuity and prove that it is equivalent to "sequential continuity". Further, we relate this to the Calculus 1 notion of continuity involving limits.
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[@tomkerruish2982]

I'll go you one better than continuity at isolated points: a function from R into R which is continuous at every irrational and discontinuous at every rational. If x is irrational, let f(x)=0; otherwise, for rational x, let f(x)=1/n, where x=m/n is in lowest terms. Since the irrationals are dense in the reals, showing discontinuity at every rational x is (relatively) easy. Showing continuity at every irrational x hinges on making your delta small enough to exclude every rational with "too small" a denominator.
For extra points, show that the set of points of continuity of any function from R into R is the intersection of countably many open sets. :D

[@JalebJay]

Was going to mention the popcorn function as well.

[@tomkerruish2982]

@@JalebJay Thank you! I never knew this function had a name. Looking it up, it appears to have a great many names. I admit to being rather taken with Conway's "Stars over Babylon." He really had a way with words.

[@Brunowsky-vj1le]

A funny consequence of the epsilon-delta definiton of continuity is that every function f:Z->R is continuous, regardless of the values it takes

[@user-uw1ut4ss2q]

It conflicts with our intuitive notion of continuity, but such anti-intuitive statement is obviously true by the rigorous mathematical definition of continuity.

[@karlluebs2605]

I think you forgot to put this video into the Real Analysis playlist. Thanks for your videos!

[@mingyarmedellin3763]

tks, from Mexico, finally someone explain this clearly
Sigue así, gracias profe saludos

[@konstantinosvelegrakis1905]

Hey mr.Penn, you should add this video to the Real Analysis Playlist!
(It may be there but even if it is,it is not placed before the "Showing a function is (dis)continuous." video!)
P.S. Your lessons help so much!

[@sea34101]

On the isolated point discussion, it reminds me this pathological case.
If instead of using the standard distance d(a,b)=|a-b|
we use the following one:
d(a,b) = 0 if a=b
d(a,b)= 1 otherwise
Then every function is continuous.
From a mathematical point of view, this distance is not very interesting (euphemism) but it helped understanding what continuity was.

[@grupiebug]

but this is actually a really good video. very nice and straight forward

[@chilledvibes99]

anyone else find these epsilon delta and all these sequence convergence proofs and definitions really confusing. I did well in maths at college but at university now and really starting to struggle with understanding this stuff even if its explained multiple times

[@sinecurve9999]

Maybe the function is only defined at rational values. Such a function would be continuous at a point (pointwise continuous?) by (3) of the theorem.

[@goodplacetostop2973]

21:39
I know I'm late but... 8pm EST is the middle of the night for me so my sleep schedule is a mess now 😂

[@davode76166]

You're late

[@guyfromkerala3577]

Happy onam.

[@elshaddai225]

If 'n'th is an odd possitive integer, prove that coefficients of the middle terms in the expansion of (x+y)^n are equal...... 👍👍

[@coxless_persian4115]

i know it's bit late, but i think you forgot to put it in the real analysis playlist

[@SzanyiAtti]

I just wanted to let you know that you forgot to put this video in the Real Analysis playlist.

[@ak12456]

I like math very much.
I learn English now, so your movie is very nice for me.
Thank you very much.

[@ajaisingh9129]

There's a typo at 5:50 delta > 0 but you accidentally ended up writing delta < 0

[@nontth5355]

is f(x)=1/x continuous in its domain?? I mean f(x) is continuous at every point on real number except x=0 and 0 is not in the domain.

[@leonardeuler6170]

Continuity is always checked on domain. So yes f:(0,a) -> R , f(x) = 1/x is continuous function.

[@murrayvonmises]

Do you climb?

[@Leyla-et9nh]

(x-1)^2x(1/sin(x-1))+5
X=1 then the limit is 5

[@arnoldvillodas4997]

Thank you 😊😊😊👍

[@AlienEntity90]

Pencil it in, perpetual.

[@christianorlandosilvaforer3451]

well ...then what we learn on calculus 1 is incorrect why they still teach that?

[@xtianthefonz1898]

what

[@grupiebug]

your breathing ....