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.

If you are going to use an ad-blocker, considering using brave and tipping me BAT!

Books I like:

Abstract Algebra:

Differential Forms:

Number Theory:

Analysis:

Calculus:

My Filming Equipment:

[Brunowsky-vjle]

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

[tomkerruish]

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

[karlluebs]

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

[konstantinosvelegrakis]

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!

[mingyarmedellin]

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

[sea]

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.

[sinecurve]

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.

[chilledvibes]

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

[goodplacetostop]

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 😂

[ak]

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

[grupiebug]

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

[ajaisingh]

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

[elshaddai]

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

[SzanyiAtti]

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

[coxless_persian]

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

[nontth]

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.

[Leyla-etnh]

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

[christianorlandosilvaforer]

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