Ramp function

The ramp function is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous definitions, for example "0 for negative inputs, output equals input for non-negative inputs". The term "ramp" can also be used for other functions obtained by scaling and shifting, and the function in this article is the unit ramp function.
In mathematics, the ramp function is also known as the positive part.
In machine learning, it is commonly known as a ReLU activation function or a rectifier in analogy to half-wave rectification in electrical engineering. In statistics it is known as a tobit model.
This function has numerous applications in mathematics and engineering, and goes by various names, depending on the context. There are differentiable variants of the ramp function.

Definitions

The ramp function may be defined analytically in several ways. Possible [|definitions] are:

It could approximated as close as desired by choosing an increasing positive value.

The ramp function has numerous [|applications] in engineering, such as in the theory of digital signal processing.
In finance, the payoff of a call option is a ramp. Horizontally flipping a ramp yields a put option, while vertically flipping corresponds to selling or being "short" an option. In finance, the shape is widely called a "hockey stick", due to the shape being similar to an ice hockey stick.
In statistics, hinge functions of multivariate adaptive regression splines are ramps, and are used to build regression models.

In the whole domain the function is non-negative, so its absolute value is itself, i.e.
and

The ramp function satisfies the differential equation:
where is the Dirac delta. This means that is a Green's function for the second derivative operator. Thus, any function,, with an integrable second derivative,, will satisfy the equation:

where is the Dirac delta.

The single-sided Laplace transform of is given as follows,

Every iterated function of the ramp mapping is itself, as