Masters in Mathematics

It looks better in analysis, when you realize that the "x" point should be an accumulation point while "L" should be an adherent point.

But srsly now, even if analysis isn't your thing, this is the minimum you have to know about it .

>B.S. in mathematics
>Never took real analysis
I just took topology and differential topology and told advisers that I self studied Rudin but never actually did. So they let me sub out differential topology for my real analysis requirement during my audit.

I imagine that if I ever try to apply to grad schools, that's going to seriously fuck me over but honestly I don't care.

creating the reals with cauchy sequences or dedekind cuts, max/min, sup/inf, compactness, and proving the calculus concepts like Intermediate value theorem, rolle, l'hospital and such. There's real analysis 1 for you.

If you're not working with analysis in gradschool you should be fine, but knowing how to play with the epsilon is quite important.

...

we learned this in first semester calculus at community college. basically the gist of it is that if you get real close over here (preimage) you get real close over there (image)

>knowing how to play with the epsilon is quite important
Yeah, that's the issue. The courses I took in undergrad were:

Calcs, diffy q's, 2 semesters of linear algebra, 2 semesters of algebra, commie algebra, topology, diffy topology, Riemannian geometry and a couple of finance classes offered through the math department.

So I never took a course where I was messing around with epsilons and inequalities outside of a couple of problems in my topology homework. I have a couple of real analysis and complex analysis textbooks (and Polya's inequalities) that I borrowed (without asking and without the intention of returning) from my uni's math lounge so maybe I'll take a look at them sometime to see what it's all about.

Well, do it for fun, you don't have to go balls deep. I'd recommend the Steven Lay analysis book, he explains it pretty well, like you're some kind of retarded or something. If you want something more sophisticated, you can try Tao's book. It's really, really good.

Also
>commie algebra

I'm laughing, what is my fucking problem? I imagined a special kind of algebra developed by the soviet union.

>commie algebra
Yeah, everyone in our class called it that. We thought we were hilarious. I think a lot of commutative algebra was developed in the USSR, though.

Consider the limit of a function that maps a domain (x) to a target (y) with an input (a) and output (b). What is the limit of f(x) at (a)?

Epsilon is just an interval of the target such that :

b - ε < b < b + ε , ε > 0

Delta is just an interval of the domain such that :

0 < | x - a | < 𝛿 , | y - b | < ε

The definition of a limit is that such a 𝛿 exists for any ε .

The idea is that no matter how small you make ε , I can choose a 𝛿 such that the sub-domain ( x - 𝛿 , x + 𝛿) maps within your sub-target ( y - ε , y + ε ) .

In other words, no matter how close you get to (b), I can get closer without using (a). Usually this is referred to as "arbitrary precision".

I don't really understand

can you work an example?

That explanation is correct but the way to visualize it is that you have a sequence and the limit exists if the terms in the sequence approach some constant value.

The epsilon delta argument says that if the limit exists, then eventually there is some term where every term after it approaches the limit to some arbitrary closeness epsilon.

It could be the first term or the millionth term, doesn't matter. The arbitrary closeness of epsilon is needed because there's no largest natural number.

did you never use pi or e or 2^0.5 or integrate\differentiate ? .
i know just the place for you senpai
youtu.be/70ouFMV8xEg?t=463

M8 if you have any topology background take a look at Baby Rudin. He starts with basic topology of metric spaces and then defines continuity (including epsilon-delta arguments) for metric spaces. Seriously, you can fix your analysis background in a few weeks.

Sure, let's consider the limit of f(x) = c , where c is some constant, at x = a .

We must find a 𝛿 such that :

0 < | x - a | < 𝛿 , | c - c | < ε

Notice that the function is simply a straight line. So no matter what 𝛿 we choose, the sub-domain ( a - 𝛿 , a + 𝛿 ) will map to exactly c . By definition, ε must be greater than | c - c | , so our mapping is within any ( c - ε , c + ε ) sub-target.

Therefor, by our definition of limit, Theorem 1.a :

lim c = c
x->a

This reads; the limit of c as x approaches a is equal to c . Notice that lim c = c at all other values of x as well.

Proving other theorems of limits is a good exercise in analysis.