realverdifunksjon

Realverdifunksjon (Norwegian for real-valued function) is a mathematical concept that denotes a function whose codomain is the real numbers. It assigns to each element of a domain X a real number, typically written as f: X -> R, where X can be any set, often a subset of the real numbers. The range of a real-valued function is a subset of R, and the function is defined by a rule or description of its values on X.

Real-valued functions are central to real analysis and appear in numerous contexts, such as calculus, probability, and mathematical modeling. Fundamental properties include continuity (informally, small changes in the input lead to small changes in the output), differentiability, and integrability. When X is a topological space, continuity is defined via the preimage of open sets in R. If X carries a measure, a function may be described as measurable if the preimage of Borel sets is measurable (with respect to the given measure).

Common special cases include continuous real-valued functions on R, bounded functions, monotone functions, and Lipschitz functions. Examples include f(x) = x^2 on R, f(x) = sin(x) on R, and piecewise-defined functions on subsets of R.

Applications of real-valued functions span modeling quantities such as position, velocity, probability densities, and cost functions in optimization. Related concepts include complex-valued functions and vector-valued functions, as well as real-valued functions of several variables, such as f: R^n -> R.