The supremum, taken over all finite partitions of a real interval, of the sums of absolute increments of a real-valued function; it measures the function's cumulative oscillation on that interval and is finite exactly when the function is of bounded variation.

Principle

Total variation organizes the behaviour of a function by accumulating all absolute changes: it ignores cancellation of signed increments and thus captures overall oscillation rather than net displacement.

Demonstration

For f(x)=x on [0,1] the total variation equals 1 because every partition sums to 1; for f(x)=|x| on [-1,1] the total variation equals 2, illustrating how monotone pieces add to the whole variation.

Misapplication

Using total variation as a substitute for net change (f(b)−f(a)) or applying the one-dimensional definition unchanged to vector-valued paths without specifying a norm can lead to incorrect conclusions about regularity.

Consequence

A real function with finite total variation admits a Jordan decomposition as the difference of two monotone increasing functions, has only jump discontinuities of bounded total size, and defines a bounded Stieltjes measure used for integration against continuous functions.

Reversal

Instead of summing absolute increments, consider the net change (signed sum) which cancels oscillations; this reversal measures displacement but loses information on roughness captured by total variation.

Boundary

Defined for real-valued functions on a closed interval (or more generally on ordered domains); it excludes naive extensions to higher-dimensional domains without choosing a vector norm or to distributions without further structure.

Semantic Tension

The phrase 'variation' is also used for statistical variance or for variations in calculus of variations; total variation here specifically denotes cumulative absolute increment, not dispersion about a mean or a functional derivative.

Synthesis

Total variation is the supremal cumulative absolute change of a real function on an interval: a quantitative, cancellation-free measure of oscillation that characterizes bounded-variation regularity and permits decomposition and measure-theoretic constructions.