The continuous-time unit step is

$$u(t)=\begin{cases}0,&t<0,\1,&t\ge0.\end{cases}$$

It represents a signal that switches on and remains on.

The continuous-time Dirac impulse $\delta(t)$ is an idealized object defined by its behavior under integration rather than as an ordinary finite-valued function. Its defining sifting property is

$$\int_{-\infty}^{\infty}x(t)\delta(t-t_0),dt=x(t_0).$$

The impulse is concentrated at one instant and has unit area in this generalized sense.

A continuous signal can be represented formally as a continuous superposition of shifted impulses:

$$x(t)=\int_{-\infty}^{\infty}x(\tau)\delta(t-\tau),d\tau.$$

This identity is the continuous-time analogue of decomposing a sequence into discrete impulses and is the starting point for continuous-time convolution.