Discrete time and continuous time

In this framework, each variable of interest is measured once at each time period.

Measurements are typically made at sequential integer values of the variable "time".

Discrete-time signals may have several origins, but can usually be classified into one of two groups:[1] In contrast, continuous time views variables as having a particular value only for an infinitesimally short amount of time.

To contrast, a discrete-time signal has a countable domain, like the natural numbers.

The electrical signals derived in proportion with the physical quantities such as temperature, pressure, sound etc.

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 point in time.

For example, is a finite duration signal but it takes an infinite value for

For some purposes, infinite singularities are acceptable as long as the signal is integrable over any finite interval (for example, the

Continuous signal may also be defined over an independent variable other than time.

For example, while economic activity actually occurs continuously, there being no moment when the economy is totally in a pause, it is only possible to measure economic activity discretely.

For this reason, published data on, for example, gross domestic product will show a sequence of quarterly values.

When one attempts to empirically explain such variables in terms of other variables and/or their own prior values, one uses time series or regression methods in which variables are indexed with a subscript indicating the time period in which the observation occurred.

For example, yt might refer to the value of income observed in unspecified time period t, y3 to the value of income observed in the third time period, etc.

Moreover, when a researcher attempts to develop a theory to explain what is observed in discrete time, often the theory itself is expressed in discrete time in order to facilitate the development of a time series or regression model.

On the other hand, it is often more mathematically tractable to construct theoretical models in continuous time, and often in areas such as physics an exact description requires the use of continuous time.

An example, known as the logistic map or logistic equation, is in which r is a parameter in the range from 2 to 4 inclusive, and x is a variable in the range from 0 to 1 inclusive whose value in period t nonlinearly affects its value in the next period, t+1.

Another example models the adjustment of a price P in response to non-zero excess demand for a product as where

For example, the adjustment of a price P in response to non-zero excess demand for a product can be modeled in continuous time as where the left side is the first derivative of the price with respect to time (that is, the rate of change of the price),

In this graphical technique, the graph appears as a sequence of horizontal steps.

Alternatively, each time period can be viewed as a detached point in time, usually at an integer value on the horizontal axis, and the measured variable is plotted as a height above that time-axis point.

The values of a variable measured in continuous time are plotted as a continuous function, since the domain of time is considered to be the entire real axis or at least some connected portion of it.