Ikkeincreasing

Ikkeincreasing is not a standard term in formal mathematics. When it appears, it is usually used informally to indicate that a function or sequence does not exhibit increasing behavior. In formal writing, researchers would more often say that the object is not increasing, not nondecreasing, or not monotone increasing, rather than adopting a fixed term such as ikkeincreasing.

Formally, for a function f defined on an ordered domain, f is not increasing if there exist points x and y with x < y such that f(x) > f(y). In other words, the function fails the monotone nondecreasing condition, which would require f(x) ≤ f(y) for all x < y.

To place ikkeincreasing in context, it is helpful to compare related monotonicity notions. Increasing (strict) means f(x) < f(y) for all x < y. Nondecreasing means f(x) ≤ f(y) for all x < y. Nonincreasing means f(x) ≥ f(y) for all x < y. A function can be nonincreasing without being characterized as not increasing in a strict sense; nonincreasing describes a monotone behavior, while not increasing describes a violation of monotone nondecreasing.

Examples illustrate the concept. The sequence 3, 2, 4, 1 is not increasing because 3 > 2 and 4 > 1 violate the nondecreasing condition. The function f(x) = sin x on [0, 2π] is not increasing over the entire interval due to its cyclic fluctuations. By contrast, the function f(x) = -x is nonincreasing and hence monotone decreasing.