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Heaviside step function

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The Heaviside step function of \scriptstyle x \, is defined as

 H(x) \equiv \frac{\sgn(x) + 1}{2} = \begin{cases}
0 & \text{if } x < 0, \\
\frac{1}{2} & \text{if } x = 0, \\
1 & \text{if } x > 0. \end{cases}

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