# Piecewise

Piecewise[{{val1,cond1},{val2,cond2},}]

represents a piecewise function with values vali in the regions defined by the conditions condi.

Piecewise[{{val1,cond1},},val]

uses default value val if none of the condi apply. The default for val is 0.

# Details • The condi are typically inequalities such as .
• The condi are evaluated in turn, until one of them is found to yield True.
• If all preceding condi yield False, then the vali corresponding to the first condi that yields True is returned as the value of the piecewise function.
• If any of the preceding condi do not literally yield False, the Piecewise function is returned in symbolic form.
• Only those vali explicitly included in the returned form are evaluated.
• Elements of the form {vali,False} are dropped, as are all elements after the first {vali,True}.
• Piecewise[conds] automatically evaluates to Piecewise[conds,0].
• Piecewise can be used in such functions as Integrate, Minimize, Reduce, DSolve, and Simplify, as well as their numeric analogs.
• Piecewise[{{v1,c1},{v2,c2},}] can be input in the form  v1 c1 v2 c2 …
. The piecewise operator can be entered as pw or \[Piecewise]. The grid of values and conditions can be constructed by first entering , then using and .
• In StandardForm and TraditionalForm, Piecewise[{{v1,c1},{v2,c2},}] is normally output using a brace, as in  v1 c1 v2 c2 …
.

# Examples

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## Basic Examples(3)

Set up a piecewise function with different pieces below and above zero:

Find the derivative of a piecewise function:

Use pw to enter and and then for each additional piecewise case:

## Scope(12)

Define a piecewise function:

Evaluate it at specific points:

Plot it:

Refine it under assumptions:

Automatic simplification of Piecewise functions:

Remove unreachable cases:

Remove False conditions:

Merge cases with the same values:

If values are not specified in a region, they are assumed to be zero:

This specifies that the default value should be 1:

Compute limits of piecewise functions:

Compute the limit in the direction of the positive imaginary axis:

Compute the series of a piecewise function:

Integrate a piecewise function:

Integration constants are chosen to make the result continuous:

Compute a definite integral of a piecewise function:

Laplace transform of a piecewise function:

Solve a piecewise differential equation:

Reduce a piecewise equation:

Integrating an implicitly piecewise integrand can give an explicit Piecewise result:

Symbolic minimization can give piecewise functions:

## Applications(1)

Compute the volume of an ellipsoid:

## Properties & Relations(11)

PiecewiseExpand converts nested piecewise functions into a single piecewise function:

Min, Max, UnitStep, and Clip are piecewise functions of real arguments:

Abs, Sign, and Arg are piecewise functions when their arguments are assumed to be real:

KroneckerDelta and DiscreteDelta are piecewise functions of complex arguments:

Boole is a piecewise function of a Boolean argument:

If, Which, and Switch can be interpreted as piecewise functions:

Convert Floor, Ceiling, Round, IntegerPart, and FractionalPart for finite ranges:

Convert Mod and Quotient when the number of cases is finite:

UnitBox and UnitTriangle are piecewise functions of real arguments:

Convert SquareWave, TriangleWave, and SawtoothWave for finite ranges:

BernsteinBasis and BSplineBasis are piecewise functions of real arguments:

## Possible Issues(1)

Derivatives are computed piece-by-piece, unless the function is univariate in a real variable:

To specify that is real, use inequalities in the first condition:

This function is discontinuous at :

Introduced in 2004
(5.1)
|
Updated in 2008
(7.0)