Step Functions

A precise guide to unit-step representations, delayed inputs, piecewise functions, and Laplace transforms.

A piecewise rule changes when the input crosses a boundary. Two differently shaped graph pieces meet near a boundary. An open endpoint and a filled endpoint distinguish a value that is excluded from one rule and included in the other.

Precise definition

The Heaviside unit step $u(t-a)$ is 0 for $t<a$ and 1 for $t>a$; its value at $t=a$ depends on convention and usually does not affect ordinary Laplace integrals. It switches a term on at time $a$. A delayed copy is $u(t-a)f(t-a)$.

Notation and mathematical language

The second shifting theorem is $\mathcal L\{u(t-a)f(t-a)\}=e^{-as}F(s)$ for $a\ge0$. A piecewise change from $f_1$ to $f_2$ at $a$ can be written $f_1(t)+u(t-a)[f_2(t)-f_1(t)]$; rewrite the bracket in powers of $t-a$ before transforming by the delay theorem.

Conceptual picture

A step representation separates a baseline from changes. Each switch contributes only after its activation time. Delayed notation preserves the shape of a response while moving its start and enforcing causality.

Fully worked example

Interpretation and application

Step inputs model switches, dosing, loads, and control commands. Real devices switch over a finite transition time, so a step is an idealization; the solution is exact for that ideal input, not a claim of physically instantaneous change.