Piecewise function
is roughly that the domain of the function can be partitioned into pieces on which the property
Sometimes the term is used in a more global sense involving triangulations; see Piecewise linear manifold.
A semicolon or comma may follow the subfunction or subdomain columns.
is rarely omitted at the start of the right column.
For bounded intervals, the number of subdomains is required to be finite, for unbounded intervals it is often only required to be locally finite.
For example, consider the piecewise definition of the absolute value function:[2]
: In order to evaluate a piecewise-defined function at a given input value, the appropriate subdomain needs to be chosen in order to select the correct sub-function—and produce the correct output value.
A piecewise-defined function is continuous on a given interval in its domain if the following conditions are met: The pictured function, for example, is piecewise-continuous throughout its subdomains, but is not continuous on the entire domain, as it contains a jump discontinuity at
The filled circle indicates that the value of the right sub-function is used in this position.
For a piecewise-defined function to be differentiable on a given interval in its domain, the following conditions have to fulfilled in addition to those for continuity above: Some sources only examine the function definition,[6][better source needed] while others acknowledge the property iff the function admits a partition into a piecewise definition that meets the conditions.
[7][8] In applied mathematical analysis, "piecewise-regular" functions have been found to be consistent with many models of the human visual system, where images are perceived at a first stage as consisting of smooth regions separated by edges (as in a cartoon);[9] a cartoon-like function is a C2 function, smooth except for the existence of discontinuity curves.
[10] In particular, shearlets have been used as a representation system to provide sparse approximations of this model class in 2D and 3D.
