Piecewise function

A function defined by different formulas or rules on different parts of its domain. Its output rule depends on which part contains the input.

Function (mathematics): A relation that assigns exactly one output to each input in its domain. A piecewise function remains a single function because every allowed input receives exactly one output.
Continuity (mathematics): A property describing when small changes in input produce small changes in output. At a boundary, matching one-sided limits determine whether adjacent pieces join continuously.
Piecewise linear function: A function whose restriction to each region of its domain is linear. Linear segments make piecewise definitions useful for approximation and practical modeling.
Step function: A function that is constant on intervals and changes value at selected points. It is a restricted piecewise function whose pieces are constant.
Domain of a function: The set of inputs for which a function is defined. The domain is divided into the regions where different rules apply.
One-sided limit: A limit approached with the input restricted to one side of a point. Each neighboring formula can have a different limit at a shared boundary.
ReLU: The function that returns the larger of its input and zero. Its two linear regions make it a standard activation in neural networks.
Spline (mathematics): A piecewise polynomial function joined under specified smoothness conditions. Splines add continuity or differentiability requirements absent from a general piecewise definition.
Interval (mathematics): A subset of the real line containing every number between any two of its members. Intervals commonly specify the input regions assigned to each formula.
Jump discontinuity: A discontinuity where finite left- and right-hand limits exist but differ. This common boundary behavior occurs when adjacent formulas approach different values.

Linked from 11 pages