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 ifthen statement, without any given statements or explanations. Although this makes each theorem considerably shorter and easier to fit onto one page than by simply copypasting 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:
 The theorems here does not explicitly define any words — look for the adjacent or embedded link in order to read about them.
 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[edit]
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 nonexistent. 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[edit]
Axioms, logically, are essentially proofs without an if statement. Thus, the following list only contains essentially then statements, which can be used freely.
Reference  Axiom 

Associative Law  
Commutative Law  
Identity Law  
Inverse Law  
Associative Law  
Commutative Law  
Identity Law  
Inverse Law  
Distributive Law  
"Equality Law" 
List of Theorems[edit]
If  Then 

a = b  f(a) = f(b) 
a = b  f^{1}(a) = f^{1}(b) 
If  Then 

the function ƒ is a constant function  
the function ƒ = x  
the function ƒ is a rational function  
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  the function is integrable over [a,b] 
the function is bounded on [a,b] and 
If  Then 

a function ƒ has a valid limit  
a function ƒ has a valid limit and it's not 0  
a function ƒ and function g has a valid limit  
a function ƒ and function g has a valid limit and


a function ƒ has a valid limit at c  the limit is unique 
If  Then 

the function ƒ is continuous at c  
the function ƒ is continuous at c and it's not 0  
the function ƒ and g are both continuous at c  
the function ƒ and g are


the function ƒ is continuous at c and the function g is continuous at g(c)  
a function ƒ is continuous on [a,b] and there exists two numbers a and b such that a < b  such that 
the function ƒ is bounded between the interval [a,b]  
a function ƒ is

such that and . 
a function ƒ is continuous on [a,b]  ƒ is integrable over [a,b] 
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:

its derivative at x is equal to 0 
Rolle's Theorem  a function ƒ is

such that 
Mean Value Theorem  a function ƒ is

such that 
Cauchy Mean Value Theorem  the functions and are

such that . 
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

ƒ(x) is a local minimum  
a function ƒ has

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. 
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]  
ƒ is integrable on [a,b]  there exists such that 
ƒ is continuous on [a,b] and ƒ is a a derivative of some function g  
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] 