1) In mathematics: a relation between a set of inputs and outputs.
2) In System Engineering: A means to impose a particular system state1.
To be useful as an Engineering abstraction, a function has to relate to the way the system’s state variables are changed (or, held constant; see also Performance Requirement). The state variables on which values are imposed to constitute the function may be either “complete” or “incomplete” with respect to the system’s Degrees of Freedom2.
The relative timing at which the control is applied is not addressed by the definition. It could be applied at either design-time, manufacturing-time, or run-time without loss of generality.
In legacy System Engineering practices, a “function” (2) is regarded as a characteristic of that system.
Compare to form and fit. See also Form, Fit, and Function, Functional Analysis, and the exegesis on Defining Functions.
Aside
Previously, I’ve phrased the definition as “…purposeful alteration (or maintenance) of the state of a system…” but the current phrasing is more concise, without loss of generality. The notion of “maintaining” a system state used to be referred to as a “dynamic state”, where the objective was to hold the system within tolerance of a particular operating point. An example would be an air conditioning system having a function allocated to maintain the temperature within specified limits.
Footnotes