# Ramp

Ramp[x]

gives x if x0 and 0 otherwise.

# Details • Ramp automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot over a subset of the reals:

Ramp is a piecewise function:

## Scope(30)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

Ramp threads over lists:

### Specific Values(5)

Values of Ramp at fixed points:

Value at zero:

Values at infinity:

Evaluate symbolically:

Find a value of x for which Ramp[x]1:

### Visualization(4)

Plot the Ramp function:

Visualize shifted Ramp functions:

Visualize the composition of Ramp with a periodic function:

Plot Ramp in three dimensions:

### Function Properties(9)

Ramp is defined for all real numbers:

It is restricted to real inputs:

Function range of Ramp:

Ramp is equivalent to the average of a number with its RealAbs:

Ramp is not an analytic function:

It has singularities, but no discontinuities:

Ramp is nondecreasing:

Ramp is not injective:

Ramp is not surjective:

Ramp is non-negative:

Ramp is convex:

### Differentiation and Integration(4)

First derivative with respect to x:

Except for the singular point , this derivative is UnitStep:

Second derivative with respect to x:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

### Integral Transforms(4)

The convolution of UnitStep with itself is equal to Ramp:

The convolution of Ramp and UnitStep gives a quadratic on the non-negative reals:

The convolution of Ramp with itself gives a cubic on the non-negative reals:

## Applications(2)

Integrate a piecewise function involving Ramp symbolically and numerically:

Solve a differential equation involving Ramp:

Plot the solution for different values of a:

## Properties & Relations(1)

Ramp is related to UnitStep: