It conflicts with our intuitive notion of continuity, but such anti-intuitive statement is obviously true by the rigorous mathematical definition of continuity.
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 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.
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!
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.
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
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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
It conflicts with our intuitive notion of continuity, but such anti-intuitive statement is obviously true by the rigorous mathematical definition of continuity.
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
Was going to mention the popcorn function as well.
@@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.
tks, from Mexico, finally someone explain this clearly
Sigue así, gracias profe saludos
I think you forgot to put this video into the Real Analysis playlist. Thanks for your videos!
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!
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.
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
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.
i know it's bit late, but i think you forgot to put it in the real analysis playlist
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 😂
You're late
but this is actually a really good video. very nice and straight forward
Happy onam.
There's a typo at 5:50 delta > 0 but you accidentally ended up writing delta < 0
If 'n'th is an odd possitive integer, prove that coefficients of the middle terms in the expansion of (x+y)^n are equal...... 👍👍
I just wanted to let you know that you forgot to put this video in the Real Analysis playlist.
I like math very much.
I learn English now, so your movie is very nice for me.
Thank you very much.
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.
Continuity is always checked on domain. So yes f:(0,a) -> R , f(x) = 1/x is continuous function.
Do you climb?
well ...then what we learn on calculus 1 is incorrect why they still teach that?
(x-1)^2x(1/sin(x-1))+5
X=1 then the limit is 5
Thank you 😊😊😊👍
Pencil it in, perpetual.
what
your breathing ....