the ladder of continuity
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- Опубликовано: 1 окт 2024
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Something we can note to 'easily' see the implications between continuities:
- For Lipschitz, we have the strongest requirement on δ, namely δ=Mε for some M. Not much of a choice at all.
- For uniform continuity, we have more options, but still restrict ourself to something only in terms of ε, ie δ=δ(ε).
- For regular continuity, we can do pretty much anything with δ. It can depend on both ε and x in any way we want.
This is nice at a definitional level, but I'd have enjoyed a bit more motivation. Why do we care about functions whose slope never exceeds some fixed value (i.e. Lipschitz)? And why do we care about function to never get too steep for very long (i.e. uniformly cont.)?
Application: if someone drives around in a car (x,y) = f(t) they will trace a curve that is lipschitz continuous.
Lipschitz continuous is really quite a strong property that creates a compact set of functions many times. Ie the set of M-LC functions on [0,1] that are between an and b is a set that is compact in many norms and is thus more easily analyzed. We often assume LC in theoretical analysis of nonparametric statistics, for example.
excuse me sir , what is M-LC and LC functions ?
A big theme in math is "does this property hold true when I take a limiting process? If not, what extra hypotheses do I need for it to be true?"
For example, when does a function have convergent fourier series, when can perimeters correctly approximate area (see pi=4 paradox), when can I exchange a limit and an integral (not pointwise convergence!), etc.
We have words like Lipschitz, pointwise, uniform, etc, to help us convey requirements for these to be true. And if you are creating an algorithm in real life which needs to be stable an analysis might require multilayered understanding: you can say this part is uniform so I can control it with a single delta, this part may blow up, etc.
Lipschitz specifically you can imagine many scenarios where you don't want a derivative to be too large: scuba divers can't ascend too fast for biological reasons, a machine may not stand a sudden temperature difference, a vehicle should not accelerate too fast, etc. If your quantity is f+g+h+e and h and e are not lipshcitz but f and g are, you know where to focus your analysis first (see if h' and e' cancel).
ruclips.net/video/t8oi3PyW3Dg/видео.htmlsi=SQDFhBLSbp-2UHTy
You should be careful with just using a subset A of R; any continuous function is uniformly continuous on a compact subset of R.
Since this condition only shows that it is uniformly continuous on that subset, I don't think much caution needs to be taken considering this is a very different condition to being uniformly continuous over R. Though I think stating that a certain continuous function is uniformly continuous on a compact domain is a mostly useless distinction unless you specifically need it to be so.
17:35
I was not old enough to not laugh on the inside at the name Lipschitz when I was in college. Apparently at 62 I am still not old enough. Great video
I still giggle at "annulus".
I was trying to think of where you might come across the Lipschitz condition in Maths. My very unreliable memory coughed-up the theory of ODEs as one possibility (proof of the existence of a solution of an ODE). And doesn't it crop up in a lot of other "fixed point" theorems?
I absolutely despised analysis, and epsilon-delta proofs but this made different continuities make so much sense! Fun fact though- Lipschitz continuity is quite a strong property- it implies a.e. differentiability of a function
I don't think it does though? Wouldn't |x| be Lipshitz continuous around 0, but not differentiable?
There are also quite exotic Hölder continuous functions which are uniformly continuous as well.
@yds6268 a.e means almost everywhere
Lipschitz is a good condition, but we can even weaken it: It's sufficient for f to be of bounded variation or absolutely continuous
@@yds6268not that "around zero, even |x| is almost everywhere differentiable! (should you not be familiar: 'a.e.' means 'almost everywhere'. I.e. everywhere except for a subset with measure zero)
ذكريات جميلة منتصف التسعينات بجامعة ابن زهر أكادير
Seeing this makes me remember an interesting theorem. Like you presented, if f is continuous it is not necessarily true that f is uniformly continuous. BUT if we restrict f to a finite interval, then it is uniformly continuous on that interval!
I'm pretty sure it must be a closed interval to work, I'm thinking of f(x)=1/x on (0,1) as a counterexample. So maybe restricted to a compact set (ie closed and bounded in R)?
In fact, in any topological space, the continuous image of a compact set is compact.
Can you also give some list of examples for what kind of functions are uniformly continuous and Lipschitz?
Like for example, we know that all elementary functions are continuous on their domain, and so we know a large class of continuous functions.
Do we have a similar list of functions, so that I can easily list out some examples for uniformly continuous and Lipschitz functions?
I was hoping for a write out of If Uniformly Continuous, then Continuous. Clearly is not always clear to the student.
how it follows |x+y| < 2|x|+1
from x+y < 2x+1
if LHS is negative ?
Learned about Lipschitz "stability" in a control theory course using the textbook "Nonlinear Systems" by Khalil. Very useful concepts.
I see how your earlier epsilon delta lesson plays into this. Today's proof would have been harder without the refresher.
Just wondering if there is a continuity of definitions of continuity and if these differ from definitions about discontinuities? 🙂
I appreciate the treatment of the connection of these topics in this way… Often it seems that topics are presented as separate facts when they are really connected by tweaks to their definitions/statements or by a hierarchy like this one 👍👍👍
From Bartle Sherbert text?
Where is absolute continuity and Hölder continuity though?
And don't forget Dini continuity 🤓
I also thought, on a metric space of a "contraction mapping" ; as far as I know, the definition's tougher than the one for Lipschitz, with the "M" constant such as : 0 < M < 1 ; the mapping must be defined over a closed segment of course ... Good luck for your excellent channel Mr. Michael Penn !
Happy New Year & Best Wishes ...
You could have included absolute continuity, as a more exotic version, and as I would have liked someone talk about it intuitively...
Nice video. I would like if you explained in the first converse proof, why you chose x and y , the way you did
Is there a function that if it's contious implies that it's Lipschitz continous?
Indian Statistical Institute, Kolkata on 16th March 😅😅😅😅
🔥
That's very nice because you tend to confuse These concepts.
There's also the holderian function
Who took the Lip out of Lipschitz?
Is continuity basically just local uniform continuity? Based on the definition.
Well, "local uniform continuity" already has an existing connotation of being uniformly continuous in some neighbourhood of each point.
In some sense yes.. but this follow by the fact that any continuous function on a compact set is uniformly continuos.
The question can be interesting if we consider function defined in space bigger than R and i'm quite sure is false
Pointwise rather than local.
Continuity doesn't necessarily imply local uniform continuity if your space isn't locally compact
Many intuitive interpretations are available, and here, one potential intuitive and motivated interpretation may be of use, that is, the notion of a "modulus of continuity", that, we express delta as a function of epsilon, and potentially of additional parameters (a terminology that is used in, for instance, Courant and John's texts "Introduction to Calculus and Analysis", volume I and II). For uniform continuity, the modulus of continuity is such that delta does not depend on the points of the domain of the continuous function given our current context, whereas for non-uniform continuous functions, we find that delta depends on the points of the domain of the continuous function. Indeed, uniform continuity implies that, whatever epsilon we are given, we can find a sufficiently small delta that satisfies the modulus of continuity everywhere in an appropriate interval regardless of the point of the interval (i.e. closed interval), whereas, for non-uniformly continuous functions, such as 1/x, the delta needs to become increasingly small for a in (0,1] where a is increasingly small for fixed epsilon, in order to "keep up" with the modulus of continuity, that the delta of the modulus of continuity of 1/x on (0,1] is dependent on the points of the interval (0,1] (compare with the linear function x on [0,1]).
Thus, there exist continuous functions that cannot be uniformly continuous locally - that is, there exist continuous functions where there does not exist an open set in which the modulus of continuity is such that delta is independent of the points of the open set.
Alternatively, one thus finds here that a certain approach to "rigorous analysis" via the use of universal quantifiers is useful. Indeed, compare the statement of continuity versus that of uniform continuity when universal quantifiers and mathematical logical notation are used - a subtle difference but easily identified notationally (see for instance Zorich's "Mathematical Analysis I").
Maybe 😏😂!!!!
Do you believe in the real numbers?
I believe in notion of labeling things by attaching things with labels.
And I think infinity is a single point stretching to infinity in every direction.
Basis: consider standing at common zero of nested supraspaces and/or subspaces and look along every axis apart from the one you choose to walk along 🙂
EDIT in light of waching video: consider standing at common zero of nested supraspaces and/or subspaces and look along every axis
EDIT in light of reflection while in reflective and stream of consciousness mode:
consider standing at common zero of nested supraspaces and/or subspaces and look along every axis as no path through N-space has been defined for you.
Thus while the axes exist there are no notions of continuity, contiguity, countably finite or finite values (labels?) in plains of intersection, if those plains exist algebraically or geometrically or ... between tuples of one or more than one of the supraspaces or subspaces of N-space, N finite or infinite
No