Heaviside Function

Evaluate the Heaviside function.

The Heaviside function is defined as

$$H(x) = \begin{cases} 1 & \textrm{if}\ x \gt 0 \\ 0 & \textrm{if}\ x \lt 0\end{cases}$$

and is discontinuous at 0. Depending on the context, the Heaviside function may be defined as a continuous function. To define the Heaviside function such that the function has rotational symmetry,

$$H(x) = \begin{cases} x & \textrm{if}\ x \gt 0 \\ \frac{1}{2} & \textrm{if}\ x = 0 \\ 0 & \textrm{if}\ x \lt 0\end{cases}$$

To define the Heaviside function as a left-continuous function,

$$H(x) = \begin{cases} x & \textrm{if}\ x \gt 0 \\ 0 & \textrm{if}\ x \leq 0\end{cases}$$

To define the Heaviside function as a right-continuous function,

$$H(x) = \begin{cases} x & \textrm{if}\ x \geq 0 \\ 0 & \textrm{if}\ x \lt 0\end{cases}$$

Usage

var heaviside = require( '@stdlib/math/base/special/heaviside' );

heaviside( x[, continuity] )

Evaluates the Heaviside function.

var v = heaviside( 3.14 );
// returns 1.0

v = heaviside( -3.14 );
// returns 0.0

v = heaviside( NaN );
// returns NaN

The continuity parameter may be one of the following values:

• 'half-maximum': if x == 0, the function returns 0.5.
• 'left-continuous': if x == 0, the function returns 0.0.
• 'right-continuous': if x == 0, the function returns 1.0.

By default, the function is discontinuous at 0.

var v = heaviside( 0.0 );
// returns NaN

To define the Heaviside function as a continuous function, set the continuity parameter.

var v = heaviside( 0.0, 'half-maximum' );
// returns 0.5

v = heaviside( 0.0, 'left-continuous' );
// returns 0.0

v = heaviside( 0.0, 'right-continuous' );
// returns 1.0

Examples

var uniform = require( '@stdlib/random/array/uniform' );
var logEachMap = require( '@stdlib/console/log-each-map' );
var heaviside = require( '@stdlib/math/base/special/heaviside' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 101, -10.0, 10.0, opts );

logEachMap( 'H(%0.4f) = %0.4f', x, heaviside );

---

See Also

• @stdlib/math/base/special/ramp: evaluate the ramp function.