Your explanation better describes the relation between limit of sequences and limit of functions. Thats the advantage of watching your videos compared to a textbook :)
In France, we have a slightly different definition for the limit of a function, we in fact allow the sequence to take the value x0 (i.e we do not have the “0
@@brightsideofmathsyes for the right and left limits we have the same definition as the one in the video, we exclude the equality with the considered point EDIT: I did a mistake in fact no the right and left limits are not same neither since we also allow for equality
2:05 Does x_0 have to be in I ? For example the function f(x)=sin(x)/x maps I=R\{0} to R. The point x_0=0 is not in I, but we still have lim_{x->0}sin(x)/x=1
Brilliant video as usual! I stop at 5:00, I look at the definition of continuity and a couple of questions come up: 1) Since any sequence is infinite, and x_0 is the limit point, the exclusion of the limit point x_0 from I in the second line follows directly. This exclusion therefore is explicitly written only for clarity. Is that correct? 2) I look at the definition and at no point do I see f(x_0) = c. Shouldn't that be written at the start perhaps?
Concerning your first question, no the exclusion of x0 doesn’t follow directly. You can consider the constant sequence equal to x0. If you do not explicitly exclude it in the definition (as it is the case in France for instance) then the limit at 0 of the function of example (a) wouldn’t exist
So this limit definition would not hold for a function which is defined pointwisely? E.g f takes values 1 for x = 1/n while value 2 for x = 1/(0.1)^n ? Then f(x_n) would not give the same limit for different x_n's. So it seems the assumption we are making is more about f than about x_n. What property is this for "f" ? Seems like continuity for me for all x's so it seems this definition says assume x is continous at every point except maybe x_0 is that right ?
In 3:46, when you say that we assume that there exists at least one sequence contained in I-{x_0} that converges to x_0 (or equivalently that x_0 is not an isolated point), you don't write it. I think it is important it to be written in the definition of limit you make, because if one just read that definition, the limit when x tends to an isolated poin of I is defined and could be any real number! Or are you supposing that I is an interval and I missed that part? Great video, thank you!
@@brightsideofmaths Ah, ok. I didn't know that isolated points were excluded implicitly just using the limit symbol. Very good videos. I'm lucky TH-cam suggested me your channel. Thank you!
So we have seen that every polynomial must be continuous. Can we say that ANY continuous function is representable with a polynomial? This is basically the idea of taylor expansion, but is it rigorously always true? If not what are the limits? I'm aware that taylor expansion is always "local" (centered on x0) so this is probably a reflection of the fact that continuousness is intrinsically a local idea.
Could you please give an example of a series? I mean a series is based on natural numbers as inputs, but inputs to a function on an interval are real numbers.
So in summary, if: 0. f is a function from I to R, where I is an open interval of R. 1. X is an element of I. 2. There exists some sequence x : N -> I\{X} that converges to X. ...then we say that, lim f(x) (x -> X) = c if and only if lim f[x(n)] (n -> ♾) = c for every sequence x : N -> I\{X} that converges to X. Furthermore, f is continuous at X if and only if c = f(X).
You really make all of this seem so easy. Keep up the good work! :D
Your explanation better describes the relation between limit of sequences and limit of functions. Thats the advantage of watching your videos compared to a textbook :)
In France, we have a slightly different definition for the limit of a function, we in fact allow the sequence to take the value x0 (i.e we do not have the “0
Yes, also a possible definition for the limit. But for the right and left limit you have the same definition?
@@brightsideofmathsyes for the right and left limits we have the same definition as the one in the video, we exclude the equality with the considered point
EDIT: I did a mistake in fact no the right and left limits are not same neither since we also allow for equality
2:05 Does x_0 have to be in I ? For example the function f(x)=sin(x)/x maps I=R\{0} to R. The point x_0=0 is not in I, but we still have lim_{x->0}sin(x)/x=1
It does not really matter. Just define the function at x_0 as well.
You should make videos on Proof Exercises from different textbooks!
I just subscribed your channel. You make amazing videos, keep up the good work!!
This is a great lecture! Thank you so much!
Brilliant video as usual!
I stop at 5:00, I look at the definition of continuity and a couple of questions come up:
1) Since any sequence is infinite, and x_0 is the limit point, the exclusion of the limit point x_0 from I in the second line follows directly. This exclusion therefore is explicitly written only for clarity. Is that correct?
2) I look at the definition and at no point do I see f(x_0) = c. Shouldn't that be written at the start perhaps?
Thanks :) Both questions will be answered with the following videos about continuity.
Concerning your first question, no the exclusion of x0 doesn’t follow directly. You can consider the constant sequence equal to x0. If you do not explicitly exclude it in the definition (as it is the case in France for instance) then the limit at 0 of the function of example (a) wouldn’t exist
So this limit definition would not hold for a function which is defined pointwisely? E.g f takes values 1 for x = 1/n while value 2 for x = 1/(0.1)^n ? Then f(x_n) would not give the same limit for different x_n's. So it seems the assumption we are making is more about f than about x_n. What property is this for "f" ? Seems like continuity for me for all x's so it seems this definition says assume x is continous at every point except maybe x_0 is that right ?
In 3:46, when you say that we assume that there exists at least one sequence contained in I-{x_0} that converges to x_0 (or equivalently that x_0 is not an isolated point), you don't write it. I think it is important it to be written in the definition of limit you make, because if one just read that definition, the limit when x tends to an isolated poin of I is defined and could be any real number! Or are you supposing that I is an interval and I missed that part? Great video, thank you!
We only use the limit symbol if x_0 is not an isolated point. Often I is an interval and then we don't have a problem at all.
@@brightsideofmaths Ah, ok. I didn't know that isolated points were excluded implicitly just using the limit symbol.
Very good videos. I'm lucky TH-cam suggested me your channel. Thank you!
damn, i learnt this material so many times yet this explanation made it seems like new subject.
This means that you still learn something new here :)
So we have seen that every polynomial must be continuous. Can we say that ANY continuous function is representable with a polynomial?
This is basically the idea of taylor expansion, but is it rigorously always true? If not what are the limits? I'm aware that taylor expansion is always "local" (centered on x0) so this is probably a reflection of the fact that continuousness is intrinsically a local idea.
No, this will not work in this way. Continuous is much more general than just being a polynomial. We will talk later about it :)
Could you please give an example of a series? I mean a series is based on natural numbers as inputs, but inputs to a function on an interval are real numbers.
Big fan
Thanks :)
So in summary, if:
0. f is a function from I to R, where I is an open interval of R.
1. X is an element of I.
2. There exists some sequence x : N -> I\{X} that converges to X.
...then we say that, lim f(x) (x -> X) = c if and only if lim f[x(n)] (n -> ♾) = c for every sequence x : N -> I\{X} that converges to X.
Furthermore, f is continuous at X if and only if c = f(X).
I missed the part when it says that I is an open interval of Rl. I thought it was an arbitrary subset of R.
1.
「上記のギフトのいずれかを選択できます」、
This is a great lecture! Thank you so much.
Thank you for your support :)