Real Analysis/List of Theorems

Below are a list of all the theorems that are covered by this wikibook. The format for each of them will not be like the theorems found throughout this wikibook however, instead they will be written as a strict if-then statement, without any given statements or explanations. Although this makes each theorem considerably shorter and easier to fit onto one page than by simply copy-pasting each proof, you will not gain the benefit of knowing how the proof is formulated nor the context for most of these theorems (which might be bad when you have no idea what each variable represents).

Note the following:

1. The theorems here does not explicitly define any words — look for the adjacent or embedded link in order to read about them.
2. For seachability reasons, this page also includes a list of properties.

And always remember that logical conditionals do not allow the converse by default!

How to Read

The theorems are divided into separate tables based on a unifying if statement. Each chart should be used like a map on where you can validly progress in your proof. The tables are divided into three rows: Reference, If, and Then. The first row is devoted to giving you, the reader, some background information for the theorem in question. It will usually be either the name of the theorem, it's immediate use for the theorem, or non-existent. The second row is what is required in order for the translation between one theorem and the next to be valid. The third row is what you can now validly assert as true, without any fear of a contradiction or an invalid statement.

If you come across any numbered lists in this page, that means that all criteria must be satisfied by default. In other words, assume that everything in that list works under AND unless stated otherwise.

Any and all normal naming conventions will be used. For example, ƒ will, by default, refer to a function — unless otherwise specified. This becomes important if certain variable names must be inferred based on context. For example, the "function" L and U for integrals actually represent the lower and upper sum, respectively, and are not necessarily the functions you are used to (so don't apply the function theorems on them!)

List of Axioms

Axioms, logically, are essentially proofs without an if statement. Thus, the following list only contains essentially then statements, which can be used freely.

List of Axioms for this Wikibook

Reference	Axiom
Associative Law	$(a+b)+c=a+(b+c)$
Commutative Law	$a+b=b+a$
Identity Law	$a+0=a$
Inverse Law	$a+(-a)=0$
Associative Law	$(a\cdot b)\cdot c=a\cdot (b\cdot c)$
Commutative Law	$a\cdot b=b\cdot a$
Identity Law	$a\cdot 1=a$
Inverse Law	$a\cdot a^{-1}=1$
Distributive Law	$a\cdot (b+c)=ab+ac$
"Equality Law"	$0\neq 1$

List of Theorems

List of Theorems and Properties for Algebra

If	Then
a = b	f(a) = f(b)
a = b	f-1(a) = f-1(b)

List of Theorems and Properties for Functions

If	Then
the function ƒ is a constant function	$f'=0$
	$\int _{a}^{b}f=c(b-a)$
the function ƒ = x	$\int _{a}^{b}f=b^{2}/2-a^{2}/2$
the function ƒ is a rational function	$f'={\dfrac {p'q-q'p}{q^{2}}}$
the function ƒ is convex over some interval I	the function -ƒ is concave over some interval I
the function ƒ is concave over some interval I	the function -ƒ is convex over some interval I
the function is bounded on [a,b] and ${\begin{aligned}&\sup {\{L(f,{\mathcal {P}})\,:\,{\mathcal {P}}{\text{ a partition of }}[a,b]\}}\\=&\inf {\{U(f,{\mathcal {P}})\,:\,{\mathcal {P}}{\text{ a partition of }}[a,b]\}}\end{aligned}}$	the function is integrable over [a,b]
the function is bounded on [a,b] and $U(f,{\mathcal {P}})-L(f,{\mathcal {P}})<\epsilon$

List of Theorems and Properties for Limits

If	Then
a function ƒ has a valid limit	$\lim _{x\rightarrow c}af(x)=a\cdot \lim _{x\rightarrow c}f(x)=aL$
a function ƒ has a valid limit and it's not 0	$\lim _{x\rightarrow c}{\frac {1}{g(x)}}={\frac {1}{M}}$
a function ƒ and function g has a valid limit	$\lim _{x\rightarrow c}{f(x)+g(x)}=L+M$
	$\lim _{x\rightarrow c}{f(x)-g(x)}=L-M$
	$\lim _{x\rightarrow c}{f(x)\cdot g(x)}=L\cdot M$
a function ƒ and function g has a valid limit and the function g is not 0 the limit of g is not 0	$\lim _{x\rightarrow c}{\frac {f(x)}{g(x)}}={\frac {L}{M}}$
a function ƒ has a valid limit at c	the limit is unique

List of Theorems and Properties for Continuity

If	Then
the function ƒ is continuous at c	$\lim _{x\rightarrow c}{\lambda \cdot g(x)}=\lambda \cdot g(c)$
the function ƒ is continuous at c and it's not 0	$\lim _{x\rightarrow c}{\left({\dfrac {1}{g}}\right)(x)}={\dfrac {1}{g(c)}}$
the function ƒ and g are both continuous at c	$\lim _{x\rightarrow c}{(f+g)(x)}=f(c)+g(c)$
	$\lim _{x\rightarrow c}{(f-g)(x)}=f(c)+g(c)$
	$\lim _{x\rightarrow c}{(f\cdot g)(x)}=f(c)\cdot g(c)$
the function ƒ and g are continuous at c g(c) ≠ 0	$\lim _{x\rightarrow c}{\left({\dfrac {f}{g}}\right)(x)}={\dfrac {f(c)}{g(c)}}$
the function g is continuous at c and the function ƒ is continuous at g(c)	$\lim _{x\rightarrow c}{f\circ g(x)}=f(g(c))$
a function ƒ is continuous on [a,b] and there exists two numbers a and b such that a < b	$\exists c\in (a,b)$ such that $f(a)<f(c)<f(b)$
	the function ƒ is bounded between the interval [a,b]
a function ƒ is continuous on [a,b] M is the upper bound for ƒ over the interval [a,b] m is the lower bound for ƒ over the interval [a,b]	$\exists c,d\in [a,b]$ such that $f(c)=M$ and $f(d)=m$.
a function ƒ is continuous on [a,b]	ƒ is integrable over [a,b]

List of Theorems and Properties for Derivatives

Reference	If	Then
Implication of Continuity	the function ƒ is differentiable at x	the function ƒ is continuous#Definition at x
Theorem Proving "Functional Analysis"	the function ƒ is: defined on (a, b) differentiable at x x is a maximum or minimum point	its derivative at x is equal to 0
Rolle's Theorem	a function ƒ is continuous on [a,b] differentiable on (a,b) f(a)=f(b)	$\exists c\in (a,b)$ such that $f'(c)=0$
Mean Value Theorem	a function ƒ is continuous on [a,b] differentiable on (a,b)	$\exists c\in (a,b)$ such that $f'(c)={\dfrac {f(a)-f(b)}{a-b}}$
Cauchy Mean Value Theorem	the functions $f(x)$ and $g(x)$ are continuous on $[a,b]$ differentiable on (a,b) are valid operands for division	$\exists c\in (a,b)$ such that ${\frac {f'(c)}{g'(c)}}={\frac {f(b)-f(a)}{g(b)-g(a)}}$.
Theorems Proving "Functional Analysis"	a function ƒ has a positive first derivative for some interval I	ƒ must also be increasing on the interval I
	a function ƒ has a negative first derivative for some interval I	ƒ must also be decreasing on the interval I
	a function ƒ has a first derivative value at 0 for some x. a positive second derivative value for some x.	ƒ(x) is a local minimum
	a function ƒ has a first derivative value at 0 for some x. a negative second derivative value for some x.	ƒ(x) is a local maximum
A Theorem Proving Intuition	a tangent line at some point (x, ƒ(x)) is between a region of convexity/concavity.	the line will be greater than the function so long as it's concave. Lesser than the function if it's convex.

List of Theorems and Properties for Integrals

If	Then
the function ƒ is integrable on [a,b] and a < c < b	ƒ is integrable on [a,c] and ƒ is integrable on [c,b]
ƒ is integrable on [a,c] and ƒ is integrable on [c,b]	the function ƒ is integrable on [a,b]
ƒ + g is integrable on [a,b]	$\int _{a}^{b}{f+g}=\int _{a}^{b}{f}+\int _{a}^{b}{g}$
ƒ is integrable on [a,b]	for all $c\in \mathbb {R}$ such that $\int _{a}^{b}{cf}=c\int _{a}^{b}{f}$
	$\int _{a}^{b}{f}=-\int _{b}^{a}{f}$
ƒ is continuous on [a,b] and ƒ is a a derivative of some function g	$\int _{a}^{b}{f}=g(b)-g(a)$
ƒ is integrable on [a,b] and ƒ is a a derivative of some function g
ƒ is continuous at c, which is in the interval [a,b]	$F'(x)={\frac {d}{dx}}\int _{a}^{x}{f}=-{\frac {d}{dx}}\int _{x}^{a}{f}=f$