A continuous signal or a continuous time signal is a varying quantity (a signal) that can be, or is expressed, as a continuous function of an independent variable, usually time.
The signal is defined over a duration, which may or may not be finite, and there is a one to one mapping of the value of the signal from the time. This basically means at any given time the signal is uniquely defined. The continuity of the time variable, in connection with the law of density of real numbers, means that the signal value can be found at any arbitrary location, t0.
A typical example of an infinite duration signal is:
f(t) = sin (t); -infinity ≤ t ≤ infinity
A finite duration counterpart of the above signal could be:
f(t) = sin (t); -pi ≤ t ≤ pi
f(t) = 0; otherwise
There are several schools of thoughts the values that the range of the function should include. One school of thought is a finite (or infinite) duration signal may or may not be finite valued. For example,
f(t) = 1/t; 0 ≤ t ≤ 1
f(t) = 0; otherwise
is a finite duration signal but it takes an infinite value for t = 0.
However, the second and possibly more prevalent school of thought is that infinity is not a limit and hence, a continuous signal must always have a finite value. This, of course, makes more sense in the case of real life continuous signals, which cannot take up infinite values in any case.
Any analogue signal is continuous by nature. Discrete signal, used in digital signal processing, can be obtained by sampling of continuous signals using a sampling function (usually a train of time-shifted impulses of the dirac delta function).
Continuous signal may also be defined over an independent variable other than time. Another very common independent variable is space and is particularly useful in image processing.