Unimodality, in the single-maximum convention, means that a real-valued function rises toward one mode or plateau and then falls away. More precisely, for a function on an interval, there is a point such that is nondecreasing to the left of and nonincreasing to its right. These inequalities are weak: a flat top is allowed, and more than one point may achieve the maximum.
For an original example,is zero throughout and decreases as one moves outward beyond that plateau. It is unimodal despite lacking a unique maximizing point. In contrast, two separated peaks with a genuine dip between them violate the rise-then-fall requirement.
A useful characterization is that a unimodal function in this convention has interval-shaped superlevel sets . For a function attaining its maximum, the converse holds too: if all those sets are intervals, a decline followed by a later rise on the same side of a maximizer would create a disconnected superlevel set. In one dimension this is the quasiconcave function viewpoint. Boyd and Vandenberghe define quasiconcavity through convex superlevel sets.[ref]
Convex Optimization by Stephen Boyd and Lieven Vandenberghe. §3.4.1, quasiconvex and quasiconcave functions
The convention must be stated. In minimization, unimodal is also used for a function that first decreases and then increases; that corresponds to a quasiconvex function and interval-shaped sublevel sets. A single minimum and a single maximum are opposite shape conditions.
Unimodality also differs from concavity. The function has derivative , so it rises up to zero and falls afterward. However,so it is not a concave function on the whole real line. Unimodality constrains the ordering of levels rather than the sign of curvature everywhere.
For a probability density, this shape defines one common meaning of a unimodal distribution. Adding symmetry around zero strengthens the conclusion: the centred interval maximizes mass for an even unimodal density lemma shows why a fixed-width observation window collects most probability when centered on the mode.
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Unimodality is a property of a function or a dataset that describes its tendency to have a single "peak" or mode. In mathematical terms, a function is unimodal if it has only one local maximum (peak) and one local minimum (trough), such that the function increases to that maximum and then decreases thereafter, or vice versa.