increasingmonotone

An increasing monotone function or sequence is one that preserves the order of its argument. More precisely, a function \(f\) defined on an interval of the real line is called increasing monotone or simply increasing if for any two points \(x\le y\) in its domain, \(f(x)\le f(y)\). When the inequality is strict, \(f\) is said to be strictly increasing. A sequence \((a_n)\) is increasing monotone if \(a_n\le a_{n+1}\) for all indices \(n\); it is strictly increasing when the inequality is strict.

Increasing monotone functions map intervals to intervals and are automatically continuous on the interior of their domain if they are bounded. In real analysis they play a central role in monotone convergence theorems: every bounded increasing sequence converges to its least upper bound, and an increasing monotone function on a closed interval attains its supremum and infimum at the endpoints.

Typical examples include the identity function \(f(x)=x\), the exponential function \(e^x\), and polynomials with nonnegative coefficients evaluated over nonnegative arguments. In discrete mathematics, nondecreasing sequences arise in the analysis of sorting algorithms and probabilistic coupling arguments.

In optimization, convex functions are automatically increasing monotone on intervals where they are defined, and many algorithms rely on the fact that an increasing monotone operator is single‑valued and has a unique fixed point in certain contexts. References for further reading include Rudin, Principles of Mathematical Analysis, and Boyd and Vandenberghe, Convex Optimization.