# UnitBox

UnitBox[x]

represents the unit box function, equal to 1 for and 0 otherwise.

UnitBox[x1,x2,]

represents the multidimensional unit box function, equal to 1 if and 0 otherwise.

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot in one dimension:

Plot in two dimensions:

UnitBox is a piecewise function:

## Scope(34)

### Numerical Evaluation(4)

Evaluate numerically:

UnitBox always returns an exact result:

Evaluate efficiently at high precision:

### Specific Values(4)

Value at zero:

Values at the points of discontinuity:

Evaluate symbolically:

Find a value of x for which UnitBox[x]=1:

### Visualization(4)

Plot the UnitBox function:

Visualize scaled UnitBox functions:

Visualize the composition of UnitBox with a periodic function:

Plot UnitBox in three dimensions:

### Function Properties(12)

Function domain of UnitBox:

It is restricted to real inputs:

Function range of UnitBox:

UnitBox is an even function:

The area under UnitBox is 1:

UnitBox has a jump discontinuity at the points :

UnitBox is not an analytic function:

It has both singularities and discontinuities:

UnitBox is neither nonincreasing nor nondecreasing:

UnitBox is not injective:

UnitBox is not surjective:

UnitBox is non-negative:

UnitBox is neither convex nor concave:

### Differentiation and Integration(6)

First derivative with respect to x:

All higher-order derivatives are the same:

First derivative with respect to z:

Compute the indefinite integral using Integrate:

Verify the anti-derivative away from the singular points:

Definite integral:

Integral over an infinite domain:

### Integral Transforms(4)

The FourierTransform of a unit box is a Sinc function:

Find the LaplaceTransform of a unit box:

The convolution of UnitBox with itself is UnitTriangle:

## Applications(2)

Integrate a piecewise function involving UnitBox symbolically and numerically:

Solve an initial value problem for the heat equation:

Specify an initial value:

Solve the initial value problem using :

Compare with the solution given by DSolveValue:

## Properties & Relations(5)

The derivative of UnitBox is a piecewise function:

The derivative of HeavisidePi is a distribution:

Convert into Piecewise:

Multidimensional unit box function equals the product of 1D functions for each argument:

UnitBox can be expressed in terms of UnitStep:

UnitBox is a special case of BSplineBasis:

Wolfram Research (2008), UnitBox, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitBox.html.

#### Text

Wolfram Research (2008), UnitBox, Wolfram Language function, https://reference.wolfram.com/language/ref/UnitBox.html.

#### CMS

Wolfram Language. 2008. "UnitBox." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/UnitBox.html.

#### APA

Wolfram Language. (2008). UnitBox. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/UnitBox.html

#### BibTeX

@misc{reference.wolfram_2024_unitbox, author="Wolfram Research", title="{UnitBox}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/UnitBox.html}", note=[Accessed: 25-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_unitbox, organization={Wolfram Research}, title={UnitBox}, year={2008}, url={https://reference.wolfram.com/language/ref/UnitBox.html}, note=[Accessed: 25-April-2024 ]}