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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 mathematics such as probability theory, numerical analysis, and elasticity.
The post-increment and post-decrement operators increase (or decrease) the value of their operand by 1, but the value of the expression is the operand's value prior to the increment (or decrement) operation. In languages where increment/decrement is not an expression (e.g., Go), only one version is needed (in the case of Go, post operators only).
In calculus, a function defined on a subset of the real numbers with real values is called monotonic if it is either entirely non-decreasing, or entirely non-increasing. [2] That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.
The theorem states that if you have an infinite matrix of non-negative real numbers , such that the rows are weakly increasing and each is bounded , where the bounds are summable < then, for each column, the non decreasing column sums , are bounded hence convergent, and the limit of the column sums is equal to the sum of the "limit column ...
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. [8] [9]
A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. [3] [4] Points where concavity changes (between concave and convex) are inflection points. [5]