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Well, well, well, back in the 1980's france used to show 12 grade student the =definition of limits through adherent points and made them prove it was equivalent to Def.1 below. (yes you read that right, 12 grade) and prove them lots of theorem about them. They had to prove the composition theorem!. But, in 1988, when they changed the mathematical books, they did a small change to the definition of limit, that today is still taught :

Def 1.

for non-french people or french people from before the change$$\lim\limits_{x\to x_0} f(x) = l \Leftrightarrow \left(0<|x-x_0|<\alpha \implies|f(x)-l|<\varepsilon\right)$$

Def 2.

Baguette definition (falsely alledged to Bourbaki) $$\lim\limits_{x\to x_0} f(x) = l \Leftrightarrow \left(|x-x_0|<\alpha \implies|f(x)-l|<\varepsilon\right)$$ Or in good ol' French

On dit que la fonction $f$ définie en $x_0$ admet pour limite $l$ en $x_0$ si et seulement si, lorsque la distance entre $x$ et $x_0$ est inférieure à $\alpha$, nombre positif aussi petit que l'on veut, la distance entre $f(x)$ et $l$ est inférieure à $\varepsilon$ ce dernier ayant la même définition que $\alpha$.

with $\alpha$ and $\epsilon$ being positive, $f$ is of course defined at point $x$ and $x_0$

The proof of the limit composition theorem becomes easier with Def 2. This is why they changed it.

What is simpler/easier with Def 2. ?

This I cannot figure out. Thanks in advance for helping me! Tom

T.D
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  • nice downvote, why? – T.D May 05 '18 at 22:17
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    Notice that definition 2 allows $|x - x_0| = 0$, meaning $x = x_0$. With the first definition, we are examining all the points around $x_0$, that are not equal to $x_0$. This is best seen with a function that is say $0$ everywhere $x \neq x_0$ and $1 $ at $x = x_0$. We could see then that with def $1$ the limit is $0$. But if we include $x_0$ we have to account for the case $f(x_0) = 1$, which would no longer make the limit $0$. Also, the second definition seems to be that of continuity, here in the states (and everywhere that is not France (?)) – rubikscube09 May 05 '18 at 22:26
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    You need to be very careful with the context; one can set up things so that the expression is only valid when all the terms are defined, so that one could say that the condition is nonsensical if $x=x_0$ but $f$ is not defined at $x_0$; or the quantification could be implicitly taken only over points in the domain. As I recall, the “french” definition is sometimes given only for accumulation points that are not in the domain, so that $|x-x_0|$ would never be actually equal to $0$. – Arturo Magidin May 05 '18 at 22:27
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    But taken at face value, definition 2 would only allow limits to exist at points where the function is defined when the function is continuous at the point. That doesn’t make it “easier” or “simpler”, it makes it different. – Arturo Magidin May 05 '18 at 22:29
  • Nice one, who said cogito ergo sum? – T.D May 05 '18 at 22:40
  • @TomDomenge the spanish philosopher Gómez Pereira, then Descartes ripped it off. – Javi May 05 '18 at 22:43
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    Relevant: https://fr.wikipedia.org/wiki/Limite_(math%C3%A9matiques)#Limite_%C3%A9point%C3%A9e – Clement C. May 05 '18 at 22:56
  • Sadly, the standard of my french education at school has been even below the level of my math education - quite an anti-achievement - ... So I would be greatful if the francophonic colleagues could translate the relevant french parts. I would actually like to know why they are using a different approach here, as Siminore's answer points to some advantages. This might actually not be about "dulling down" school standards but constitute a serious albeit different approach. However if french quotes are thrown around I will have to capitulate... – mol3574710n0fN074710n May 05 '18 at 23:12
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    @mol3574710n0fN074710n the article referenced above states that for functions, the "English" standard notion of limit corresponds to the French notion of pointed limit, while the French notion of limit corresponds to what is written in the OP. Various relations between these notions (e.g., that they are equivalent if $x_0$ is not a point of the domain) are then discussed. – Clement C. May 05 '18 at 23:14
  • There also is a discussion w.r.t. to that distinction (partly in English, mostly in French) in the Wikipedia:Talk page.. – Clement C. May 05 '18 at 23:17
  • @Clement C. -- Thank you, I now feel included in the discussion ;) - Nothing about the motivation of / ideas behind that different concept though? Maybe it is purely for historical reasons? – mol3574710n0fN074710n May 05 '18 at 23:19
  • @mol3574710n0fN074710n I am not sure, but if you want to go down the rabbit hole of links and references, following the refs from that Wikipedia page: http://mymathforum.com/real-analysis/646-limit-function-metric-space.html "Meaning to Bourbaki's followers, a function which, for instance, has a limit at point x0 in R and which is defined at this point will automatically be continuous. This definition has the advantage of making the formulation of a lot of theorems simpler (especially those involving the composition operator)." – Clement C. May 05 '18 at 23:22
  • Ah, at the wiki-talk page they say it is indeed about compositions of functions! (And the french they write in talk is way more accessible to me than the rather elaborate language of the wiki main pages!) – mol3574710n0fN074710n May 05 '18 at 23:24
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    Tom Domenege, @ClementC.'s comment shows that you are completely wrong in your sentence "for non-french people or french people from before the change". You are comparing the "non french limit" with the french "Limite pointée" while a fair comparison should compare the "non french limit" with the french "Limite épointée", which are exactly the same thing. The thing you are missing is that in France there are two concepts of limit: one particular ("pointée"), and one general ("épointée") which conincides with the "non french". – Pedro May 05 '18 at 23:25
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    I went to lycée and university in France (very much post 1988), and Def. 2 is definitely not taught as the definition of the limit. I don't think any of my professors or fellows students would claim that the indicator function of ${0}$ has no limit at $0$. Maybe if you go digging through lycée textbooks you'll find this definition, but if so, nobody seems to remember it by the time they get to university. In any case, the two definitions are not equivalent, so asking which proofs are made easier seems like a meaningless question. – Jack M May 05 '18 at 23:25
  • @Clement C. -- Thanks, that is indeed what I was looking for. I somehow have to fill the time between Matlab's calculation runs, after all XD. – mol3574710n0fN074710n May 05 '18 at 23:26
  • @Pedro (in case you read it -- forget my previous, now deleted comment. Mistranslation on my side) To add to the body of evidence (in French): http://www.les-mathematiques.net/phorum/read.php?18,1424120,1426100 – Clement C. May 05 '18 at 23:29

3 Answers3

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The absolute value can now become $0$. I believe that might imply that $f$ has to be defined in $x_0$, but that would be a matter of context...

If $f$ has to be defined in $x_0$ there will be big trouble with extending a continuous function's domain by using the limit (see sinc-function).

But it is soothing that at least the french still learn some math at school. I should migrate...

EDIT and remark: (for future visitors)

An assumption of $x_0$ being in the domain of $f$ may or may not be made at school, this I cannot assess.

From Siminore's answer and from the comments to the question, linking the french wikipedia, it has however emerged that the specific definition of a limit where $x_0$ is indeed part of the domain of $f$ can be helpful with respect to proving statements about the limit of the composition of several functions.

mol3574710n0fN074710n
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I have met a colleague in Brescia who teaches limits the french way. In his notes he says that the only reason to prefer the french definition is that the limit of compositions becomes easier. I do not believe that there are other reasons, actually.

Siminore
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What is simpler/easier with Def 2. ?

Maybe the relevant issue is pedagogic.

I suspect that, for 12 grade students (which is the context of your question), "functions defined on some open interval that contains $x_0$" are easier to handle (conceptually) than "functions defined on some open interval that contains $x_0$, except possibly at $x_0$ itself".

The biggest "problem" of Definition 2 is that, according to it, a function cannot have a limit at a discontinuity point of its domain. But this is not a real problem if our analisys considers only continuous functions (which would be reasonable for 12 grade students).

Pedro
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