A function assigns exactly one output to each input in its domain.
Key knowledge
A function f:A→B assigns exactly one element of B to each element of A. A is the domain, B the codomain, and the range consists of outputs actually attained.
Values and equations
f : A → B
f
Function
A
Domain
B
Codomain containing every output
Conditions and scope
This record uses the total-function convention; a partial function may be undefined on some inputs.
Declare the domain and codomain; different inputs may share an output.
An undefined expression at an input does not give a function value there.
Sources · 4
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
2026-10-10 · Compared elementary total-function notation with MIT's explicit distinction between partial and total functions.
Foundational review
2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.
Sources from other places
Agreement within this scope
Total functions give one output per input, distinct from partial functions.
The source explicitly preserves the total-function convention and domain/codomain.
A function f:A→B assigns exactly one element of B to each element of A. A is the domain, B the codomain, and the range consists of outputs actually attained.
A real linear map T:ℝⁿ→ℝᵐ satisfies T(αu+βv)=αT(u)+βT(v). After choosing input and output bases, it is represented by an m×n matrix A: the output coordinates are y=Ax, with yᵢ=Σⱼ₌₁ⁿaᵢⱼxⱼ.
The limit of f(x) as x approaches a is L if every positive output tolerance ε admits a positive input tolerance δ such that 0 < |x−a| < δ implies |f(x)−L| < ε.
A set is countably infinite if it has a bijection with the natural numbers. Here, countable means finite or countably infinite, equivalently admitting an injection into N.
Interpolation chooses p with p(tᵢ)=yᵢ at specified distinct nodes. Global high-degree polynomials can behave poorly between nodes; piecewise low-degree interpolation limits degree locally, and node choice affects conditioning.
A map between measurable spaces is measurable when every measurable target set has a measurable inverse image. This depends on the sigma-algebras; measures are not needed to define the map's measurability.
A self-map T of a nonempty complete metric space with d(Tx,Ty)≤k d(x,y), 0≤k<1, has one fixed point. Iteration xₙ₊₁=T(xₙ) converges to it from any starting point.
A DFA consists of finite states Q, finite alphabet Σ, total transition δ:Q×Σ→Q, start q₀ and accepting states F. It reads a finite string once, accepts when the final state lies in F and recognizes a regular language.
For eventually nonnegative costs f,g, f=Θ(g) means c₁g(n)≤f(n)≤c₂g(n) for all sufficiently large n and constants c₁,c₂>0. The lower inequality alone defines f=Ω(g), and the upper defines f=O(g).
An invertible affine transformation of ℝⁿ has form T(x)=Ax+b with A invertible. It preserves affine combinations whose weights sum to one, collinearity, parallelism and the signed parameter along a line. General affine transformations need not preserve Euclidean lengths or angles.
The composition (f∘g)(x) is f(g(x)), defined when x belongs to g's domain and g(x) belongs to f's domain. A one-to-one function has an inverse on its actual range, reversing its input-output assignment.
Composition requires compatible domains and inversion uses a one-to-one input-output assignment.