# Function Theory: TOPIC

* Function Theory
* Function definition
  * function definition
  * function as a set
  * function as a relation
  * function as a special relation
  * left-unique relation
  * right-serial relation
  * function as a rule
  * function as a process
  * function is a set
  * function is a set of ordered pairs
  * as a graph
  * as a rule (procedure)
* Function components
  * domain
  * codomain
  * range
  * image
  * pre-image
* Function properties
  * injectivity
  * surjectivity
  * bijectivity
  * totality
  * total function
  * partial function
  * primitive function
  * one-to-one function
  * onto function
  * surjection
  * injection
  * bijection
  * one-to-one function
  * one-to-one correspondence
  * map
  * mapping
  * transformation
  * morphism
* Function types
  * total
  * partial
  * identity function
  * invertible function
  * constant function
* Function concepts
  * fixpoint
* Function in programming
  * named function
  * anonymous function, x ↦ f (x)
* Functions in computability theory
  * Turing-computable function
  * μ-recursive functions
  * Computable function
  * PRF
  * GRF
* Functions: alt
  * map, mapping
  * transformation
  * morphism
  * arrow
* Morphisms
  * homomorphism
  * homeomorphism
  * endomorphism
  * hylomorphism
  * catamorphism
* Examples by domain and codomain
  * X → 𝔹, 𝔹 → X, 𝔹ⁿ → 𝔹
  * X → ℕ, ℕ → X
  * X → ℤ, ℤ → X
  * X → ℝ, ℝ → X, ℝⁿ → X
  * X → ℂ, ℂ → X, ℂⁿ → X
* Constructions  
  * restriction
  * composition
  * inverse
* Generalizations  
  * partial function
  * multivalued function
  * implicit function
* Lambda Calculus formalisam of functions function abstraction function application λ-calculus lambda calculus function abstraction function application alpha equivalence beta conversion eta conversion
* function notation function notation arrow notation index notation dot notation specialized notations
* function specification by listing function values by a formula inverse function implicit function using differential calculus by recurrence relation
* representing a function set of ordered pairs as a formula graphs and plots table bar chart
  * standard functions
  * composition
  * image and preimage
  * injective function
  * surjective function
  * bijective function
  * restriction and extension
  * multivariate function
  * calculus
    * real function
    * vector-valued function
  * misc
    * function space
    * multi-valued functions
    * in the foundations of mathematics and set theory
    * generalizations