Ramp

The ramp function: x for x >= 0, 0 for x < 0.

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Ramp(x)max(x, 0).

Typed signatures

Type	Package	Applies
(real) -> number	analytic	always

Domain: Elementary functions

Details

• Ramp(x) = x * UnitStep(x) -- continuous, so Ramp(0) = 0 needs no special case.
• D(Ramp(x), x) = Piecewise({{0, x < 0}, {1, x > 0}}, Indeterminate) -- NOT UnitStep(x), which would be defined (as 1) at x = 0; Ramp's own slope there is undefined.

Plot

Examples

Basic #

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In$\mathrm{Ramp}(-2)$Out$0$

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In$\mathrm{Ramp}(3)$Out$3$

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In$\mathrm{Ramp}(0)$Out$0$

Scope #

An exact irrational input at or above 0 is handed back untouched

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In$\mathrm{Ramp}(π)$Out$π$

Known value · Ramp(v) = v for v ≥ 0, exactly -- π is never rounded

An exact irrational input below 0

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In$\mathrm{Ramp}(-π)$Out$0$

Known value · Ramp(v) = 0 for v < 0

Properties #

Ramp(x) = x * UnitStep(x)

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In$\mathrm{Ramp}(5)=5\mathrm{UnitStep}(5)$Out$\mathrm{True}$

0 or 1 away from the kink, undefined at x = 0

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In$\frac{\mathrm{d}\mathrm{Ramp}(x)}{\mathrm{d}x}$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack0, x\lt0\bigr\rbrack, \bigl\lbrack1, 0\lt x\bigr\rbrack\bigr\rbrack, \operatorname{Indeterminate})$

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