This function is applied in engineering (e.g., in the theory of DSP). The name ramp function can be derived by the look of its graph.
The ramp function () may be defined analytically in several ways. Possible definitions are:
- The mean of a straight line with unity gradient and its modulus:
this can be derived by noting the following definition of ,
for which and
- The Heaviside step function multiplied by a straight line with unity gradient:
- The convolution of the Heaviside step function with itself:
- The integral of the Heaviside step function:
- Proof: by the mean of definition  it is non-negative in the I. quarter, and zero in the II.; so everywhere it is non-negative.
Its derivative is the Heaviside function:
From this property definition . goes.
The single-sided Laplace transform of is given as follows,
Every iterated function of the ramp mapping is itself, as
We applied the non-negative property.