 Heaviside step function

The Heaviside step function switches from zero to one at its threshold. With threshold zero,The value at zero is a convention, commonly zero, one half or one; changing this single value changes no ordinary integral. The delayed signal $H(t-a)f(t-a)$ is zero before time $a$ and then follows the original signal from its beginning. This is the switch used by the Laplace transform shift theorem.

As a distribution, its derivative is the Dirac delta function: $H^{\prime }=\delta$. Indeed, against a smooth compactly supported test function $\phi$, one has $⟨H^{\prime },\phi ⟩=-{\int }_0^{\infty }{\phi }^{\prime }(t)dt=\phi (0)$. Ordinary pointwise differentiation away from zero would miss this concentrated jump.

The Heaviside step function, often denoted as \( H(t) \) or \( u(t) \), is a piecewise function that plays a significant role in various branches of mathematics and engineering, particularly in control theory and signal processing.