In many instances, the data source itself determines whether information is represented in a continuous or discrete fashion. For example, digital information, such as a file stored on a disk, is represented as a series of 1s and 0s. This information has no value between these data points and therefore must be represented as a discrete data type. Continuous data however, such as the sine wave generated on an oscilloscope, possesses a value at all points along its domain that is dependent upon what point of the domain is being examined.
Continuous data is reflected as a graph where all data points have a meaningful value. An example of this would be the trigonometric sine wave. Discrete data, however, is represented as only points, usually above whole number integers, on a graph. While there are sometimes lines connecting these points, these lines do not represent actual values at those points along the domain and rather serve as trends or average slope lines between changing domain values.
Continuous functions, equations that represent continuous data, are the primary tools of mathematics. These functions allow for tonicity to be determined as well as other important information such as slope and inherent value. Discrete functions, usually first encountered in the form of infinite series, are commonly used as approximations when a continuous function cannot be adequately discerned. Discrete functions also allow for non-continuous data sources, such as the average temperature per day, to be analyzed and meaningful information derived from them.
Continuous functions lend themselves to higher-level manipulation in mathematics. For example, one of the prerequisites of the operations of differentiation and integration is that a function be continuous. Continuous data is also easily derived in natural phenomena. For example, very few natural occurrences, such as temperature change, time and sound, occur in a discrete fashion. Discrete data is often how natural phenomena is recorded and allows approximations, such as through Taylor and Maclaurin series, for continuous data. A prime example of this is the approximation of the sine function. Calculators use a Maclaurin series to approximate a valid answer for this function as digital devices are unable to process continuous data.