Ramp function

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

${\displaystyle \forall x\in \mathbb {R} :R(x)\geq 0}$

and

${\displaystyle \left|R(x)\right|=R(x)}$

• Proof: by the mean of definition 2, it is non-negative in the first quarter, and zero in the second; so everywhere it is non-negative.

Its derivative is the Heaviside function:

${\displaystyle R'(x)=H(x)\quad {\mbox{for }}x\neq 0.}$

The ramp function satisfies the differential equation:

${\displaystyle {\frac {d^{2}}{dx^{2}}}R(x-x_{0})=\delta (x-x_{0}),}$

where δ(x) is the Dirac delta. This means that R(x) is a Green's function for the second derivative operator. Thus, any function, f(x), with an integrable second derivative, f″(x), will satisfy the equation:

${\displaystyle f(x)=f(a)+(x-a)f'(a)+\int _{a}^{b}R(x-s)f''(s)\,ds\quad {\mbox{for }}a<x<b.}$

${\displaystyle {\mathcal {F}}{\big \{}R(x){\big \}}(f)=\int _{-\infty }^{\infty }R(x)e^{-2\pi ifx}\,dx={\frac {i\delta '(f)}{4\pi }}-{\frac {1}{4\pi ^{2}f^{2}}},}$

where δ(x) is the Dirac delta (in this formula, its derivative appears).

The single-sided Laplace transform of R(x) is given as follows,[2]

${\displaystyle {\mathcal {L}}{\big \{}R(x){\big \}}(s)=\int _{0}^{\infty }e^{-sx}R(x)dx={\frac {1}{s^{2}}}.}$

Every iterated function of the ramp mapping is itself, as

${\displaystyle R{\big (}R(x){\big )}=R(x).}$

• Proof:

${\displaystyle R{\big (}R(x){\big )}:={\frac {R(x)+|R(x)|}{2}}={\frac {R(x)+R(x)}{2}}=R(x).}$

This applies the non-negative property.