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→Formulation: Added the piece-wise definition also to the other two kinds of Heaviside function |
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For the alternative convention that {{math|''H''(0) {{=}} {{sfrac|1|2}}}}, it may be expressed as:
* a [[piecewise function]]: <math display="block">H(x) := \begin{cases} 1, & x > 0 \\ \frac{1}{2}, & x = 0 \\ 0, & x < 0 \end{cases}</math>
* a [[linear transformation]] of the [[sign function]], <math display="block">H(x) := \frac{1}{2} \left(\mbox{sgn}\, x + 1\right)</math>
* the [[arithmetic mean]] of two [[Iverson bracket]]s, <math display="block">H(x) := \frac{[x\geq 0] + [x>0]}{2}</math>
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Other definitions which are undefined at {{math|''H''(0)}} include:
* a [[piecewise function]]: <math display="block">H(x) := \begin{cases} 1, & x > 0 \\ 0, & x < 0 \end{cases}</math>
* the derivative of the [[ramp function]]: <math display="block">H(x) := \frac{d}{dx} \max \{ x, 0 \}\quad \mbox{for } x \ne 0</math>
* in terms of the [[absolute value]] function as
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