definitions/theorems

I don't know OP why not crack open a real analysis textbook and find out.

Definition of the natural numbers, rationals, reals
Definitions of limits, deratives, anti-deratives
A proper understanding of Eulers identity

>>8567285
I'd just like to interject for a moment. What you're referring to as calculus, is in fact, real analysis, or as I've recently taken to calling it, Undefined control sequence \leqslant-analysis. Calculus is not a branch of mathematics unto itself, but rather another application of a fully functioning analysis made useful by topology, measure theory and vital R-related properties comprising a full number field as defined by pure mathematics.

Many mathematics students and professors use applications of real analysis every day, without realizing it. Through a peculiar turn of events, the application of real analysis which is widely used today is often called "Calculus", and many of its users are not aware that it is merely a part of real analysis, developed by the Nicolas Bourbaki group.

There really is a calculus, and these people are using it, but it is just a part of the field they use. Calculus is the computation process: the set of rules and formulae that allow the mathematical mind to derive numerical formulae from other numerical formulae. The computation process is an essential part of a branch of mathematics, but useless by itself; it can only function in the context of a complete number field. Calculus is normally used in combination with the real number field, its topology and its measured space: the whole system is basically real numbers with analytical methods and properties added, or real analysis. All the so-called calculus problems are really problems of real analysis.

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110KB, 1342x436px

>>8568490

definition of a metric/topological space.
definition of a function
definition of a limit (of a function or a sequence)
definition of continuity
definition of connected and compact sets
derivatives, its intermediate value property
Rolle, Lagrange, Cauchy intermediate value theorems.
uniform continuity and uniform convergence.
and a lot more of stuff probably

>>8568561
are a lot of these topology though?
also is spivak's calculus good for these definitions and theorems?

>>8568579
there is nothing in that list that is "too topological" or whatever. just trivial topology
haven't read spivak. grab an analysis book

>>8568659
i read some of rudin's principles, but many of his definitions are in terms of metric spaces. is that normal or

>>8568768
if you understand the spivak "dumbed down" version of things like limits and sequences in R and whatever it would probably take you 20 seconds to understand the more general version in regular analysis books like rudin. the extension is very trivial, essentially replacing wherever you would see the absolute value symbol with an arbitrary distance function

>>8568768
in fact the difficulty gap between spivak and rudin is actually very small. one could certainly do rudin without first doing spivak. the difficulty in exercises are very similar. some of the problems are even completely identical like proving the limit of arithmetic mean of a sequence is the same as the sequential limit if it exists and such things

>>8568793
>>8568791
interesting okay cool will consider

>>8567285
>L'hôpital's rule
kys

>>8567285
Riemann rearrangement theorem is good stuff. Blew my mind when I first read it.

>>8568810
whats wrong with l'hopitals rule