Function of a real variable

A function of a real variable maps each input from a subset of the real numbers to exactly one real output. Its domain lies on the real line, and its values lie on the real line.

Connect

Domain of a function: The set of inputs for which a function is defined. It specifies which real numbers a function can accept.
Function composition: The operation that applies one function to the output of another. Composing real-variable functions builds more complex input-output rules.
Optimization: The study of finding inputs that minimize or maximize an objective function. A real-valued objective assigns a quantity to each candidate input.
Function of a complex variable: A function whose input and output are complex numbers. Complex inputs and outputs permit behavior unavailable to real-variable functions.
Codomain: The declared set of possible outputs of a function. It distinguishes the target set from the values the function actually attains.
Inverse function: A function that reverses another function’s input-output mapping on suitable domains. An inverse recovers an input when the original function is one-to-one.
Ordinary differential equation: An equation relating an unknown function of one variable to its derivatives. Its solutions are functions whose rates of change satisfy prescribed rules.
Multivalued function: A relation that assigns multiple possible outputs to at least one input. A real-variable function must assign exactly one output to each input in its domain.
Graph of a function: The set of ordered input-output pairs represented as points in a coordinate plane. It turns the function’s real inputs and outputs into a geometric picture.
Implicit function theorem: A theorem giving conditions under which an equation locally defines one variable as a differentiable function of others. It shows when a relation involving real variables determines a local function.