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The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics. For example, in classical analysis they occur in the proof of the positivity of integrals involving Bessel functions or the positivity of Cesàro means of certain Jacobi series. [6] Such functions occur in other areas of ...
By the inverse function theorem, a continuous function of a single variable (where ) is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima). For example, the function. is invertible, since the derivative f′(x) = 3x2 + 1 is always positive.
A monotonically non-increasing function. Figure 3. A function that is not monotonic. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [1][2][3] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
Then f is a non-decreasing function on [a, b], which is continuous except for jump discontinuities at x n for n ≥ 1. In the case of finitely many jump discontinuities, f is a step function . The examples above are generalised step functions; they are very special cases of what are called jump functions or saltus-functions.
Derivative test. In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is ...
The function is a symmetric and decreasing function whose level sets have the same measure as the level sets of that is, If is a function in then. The Hardy–Littlewood inequality holds, that is, Further, the Pólya–Szegő inequality holds. This says that if and if then. The symmetric decreasing rearrangement is order preserving and ...