Functiononscalar

Functiononscalar is a term used to describe a function whose input is a scalar value. In mathematics, a scalar is a single numeric value such as a real number, an integer, or a complex number. A function on a scalar thus maps a single input to an output, which may itself be a scalar or a more complex object depending on the context.

In the most common mathematical setting, a real-valued function on scalars has the form f: R -> R, with examples including f(x) = x^2, f(x) = sin(x), and f(x) = e^x. Domain restrictions are common, such as limiting x to an interval. By contrast, when the input domain is a vector, such as f: R^n -> R, the function is not classified as a function on scalars, but rather as a multivariate or vector-valued function.

In computer science and programming, a function on a scalar refers to a function that accepts scalar arguments and typically returns a scalar result, as opposed to functions that operate on arrays, matrices, or other composite data structures. Examples include arithmetic and transcendental operations like sqrt, log, and abs. Some languages offer vectorization, applying a scalar function to each element of a collection, while others require explicit iteration.

A related concept is a scalar field, which is a function that assigns a real number to every point in a space, such as a manifold. Although called a scalar function as well, a scalar field differs from a simple function on a scalar input because its domain is a space of locations rather than a one-dimensional scalar domain.

See also: scalar field, vector-valued function, multivariable function, real-valued function.