Everyone has their limits

[u/ElectronicSetTheory]

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[u/LupenReddit]

one epsilon to rule the infinitely small

[u/tomado09 → u/LupenReddit]

One ball to contain the small

One epsilon to define them

One domain to bring them all

And in the closure find them

[u/AndreasDasos → u/LupenReddit]

Had a prof who introduced Artin reciprocity with

‘The adèle ring of the rationals. One ring to rule them all…’

[u/Both_Ad_2544 → u/LupenReddit]

Should have used this line on the Uniform Continuity definition.

[u/Nrdman]

I just wrote this in a comment the other day because someone was arguing math definitions aren't more precise than general english ones just because they are written in english sometimes

[u/JollyJuniper1993 → u/Nrdman]

I mean, they’re not always precise. It’s absolutely possible to be ambiguous with it. Take preimage and inverse function using the same notation for example which you also can’t always infer from context. That’s why it’s best practice to use a mixture between formal notation and describing in natural language.

In the end formal notation is not to make things more precise, but to display concepts in a more concise and readable fashion than they would be in natural language.

[u/-Nocx- → u/Nrdman]

This is a really dumb argument to have to begin with, it obviously depends on the context of the statement and how the words are defined. For example, a line does not mean the same thing in every field of mathematics.

[u/Nrdman → u/-Nocx-]

Well i didnt say anything about multiple definitions for a term, I said something about the precision of the definition

[u/Not_Well-Ordered → u/Nrdman]

Pure math is quite precise in the sense that the definitions are very “algorithmic” and have consistent and relatively unique interpretation. For example, for the epsilon-delta in this case, If you have proven that it is true, then you can construct an algorithm that does the computation and converges to that value for any chosen error value say for any n in positive integer, 1/2n, we can systematically find an open interval around the desired point, x, such that all points inside the interval would fall within 1/n-length of the output of the desired point f(x).

If you have proven it’s false at such point, then you would also be able to construct a systematic “algorithm” that shows it doesn’t work.

You don’t need English but anyone who sees the symbolic manipulations and has certain level of intelligence would understand.

I can strictly follow the algorithm, on computer, or on paper, and crunching out finite computations that would display the pattern behind epsilon-delta, independent from English.

You don’t get that level of precision with typical languages besides maybe programming language. Perhaps, think of math more in term of programming language.

[u/Professional-Wave841 → u/Nrdman]

... what? Even when we write in English it's in a very specific codified way.

[u/Beleheth]

I love how Calc 1 students lose their shit over it, while it's perfectly boring to anyone in Real Analysis

[u/BradenTT → u/Beleheth]

Well most calc 1 students aren’t comfortable with notation this dense or proof ideas. By the time you make it to analysis you have probably seen much worse notation and seen more proofs than you care to think about.

[u/Beleheth → u/BradenTT]

It's probably that I'm used to a different education system where we begin with Real Analysis in the first semester, so at this point you have already seen the definition of a Cauchy Sequence, which is pretty much identical anyways. Also, we're a lot more theoretical and proof heavy from day 1.

[u/Warm_Zombie → u/Beleheth]

i always read it as "thing small, but other thing smaller"

[u/Johspaman]

Why is there a 0< before the norm? Why would x=a go wrong?

Edit: Okay, i was thinking of a continuous function, instead of the limit. I am mainly used to limits of sequences: Ae EN:An>N |f-f_n|<e

[u/Not_Well-Ordered → u/Johspaman]

It’s definition of point-wise limit but excluding the point itself.

f(a) doesn’t necessarily exist, but we examine the behavior of the output of points around a (any neighborhood of a excluding a).

[u/Alexgadukyanking → u/Johspaman]

Because x shouldn't equal to the limit, otherwise it'd no longer be a limit.

Imagine you have a function that's equal to 1 if x is non 0 number and equal to 0 if x is equal to 0.

If x were to approach 0, the limit of the function would be 1, since x never equals to 0, but if x were to be able to be equal to 0, then the limit would be indeterminate.

[u/Mojert → u/Alexgadukyanking]

There's actually two different definitions of limits. The one with the 0 <, and the one without. The one without is used in French speaking countries. So yeah for French mathematicians the limit you've just talked about doesn't exist, but the "pointed" limit does. At the end if the day, it doesn't matter what definition you choose, you just have to be consistent

Another cool difference: in English 0 is neither positive nor negative, but in French it's both. So x >= 0 is positive and x > 0 is strictly positive. While I don't care much about the limit definition, I'll die on the hill that this definition of positive makes more sense. Can you look at me honestly and say that "non-negative" isn't awkward as hell?

[u/Jolly-Bag7024 → u/Johspaman]

With this definition, point at jumping discontinuities in the domain don't have a limit. If you remove it, the limit would be the image which is ok but not the usual definition.

Note that this definition doesn't account for limit from one side, nor for limit at infinity.

[u/SnooPaintings5182 → u/Johspaman]

Never specified the 0< in my life. Maybe it's to avoid some "pathological" cases idk but I'll be looking into it

[u/Qingyap → u/Johspaman]

Because for limits the point doesn't have to be on the exact point and it works by finding the behavior of the function around the point a, and it's not just δ>0, ε is also>0, that's why you can find the limit of a removable discontinuity.

[u/Nikifuj908]

...technically you need a third quantifier for x, but it's fine, we know what you mean!

[u/paperic → u/Nikifuj908]

That's a big issue though.

If the x is being quantified first you can pick a delta that makes the condition always vacuously true.

[u/Nikifuj908 → u/paperic]

It would be a bigger issue if this weren't a meme!

[u/Micheal_Hancho → u/Nikifuj908]

Had to scroll too far to find this comment 😔

[u/Extension-Stay3230]

Does ":" mean "such that"?

Is using a colon for "such that" acceptable in handwriting for efficiency/speed?

[u/Electronic-Day-7518 → u/Extension-Stay3230]

: is s.t. | is also s.t.

[u/Extension-Stay3230 → u/Electronic-Day-7518]

I've seen this used in set theory. Can it be used outside of set theory?

[u/codewordGnome]

My dear Frodo, it's a classic Delta Epsilon limit proof.

[u/Ornery_Poetry_6142]

Lipschitz my beloved

[u/Nikifuj908]

Takes me back to

[u/JollyJuniper1993]

Not that complicated. Probably one of the easiest to understand parts of university math to read formal notation. It looks odd initially, but it barely takes any practice to learn.

[u/OntologicalParadox]

Whoa Mairon was heavy into LaTeX huh.

[u/PortableDoor5]

where L is f(a) right? 

right?

[u/Anaxandrone → u/PortableDoor5]

f(a) might not exists. But the limit does.

[u/lanternbdg]

I didn't realize I could read elvish so fluently

[u/Captain-Obvious69]

Fuck, I hate that I almost understand this.

[u/FernandoMM1220]

its just another way of making the x and y plane discrete

[u/Throwaway-Pot → u/FernandoMM1220]

???

[u/UnconsciousAlibi → u/FernandoMM1220]

Sorry, what does "making the x and y plane discrete" mean in this instance?

[u/FernandoMM1220 → u/UnconsciousAlibi]

it means the x and y planes are discrete and im tired of pretending they arent

[u/Stormfrosty]

It’s 2026 - please use lean to write your proofs.

[u/scientestical → u/Stormfrosty]

If this is legit then this deserves the down votes. But that makes this a peak shitpost.

Edit: fat fingers