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In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions.
The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the non-monotonic function shown in figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor ...
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...
Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1.
As explained in Riesz & Sz.-Nagy (1990), every non-decreasing non-negative function F can be decomposed uniquely as a sum of a jump function f and a continuous monotone function g: the jump function f is constructed by using the jump data of the original monotone function F and it is easy to check that g = F − f is continuous and monotone. [10]
See Theorem 12.1 on page 157 for the monotone increasing case. Rudin, Walter R. (1976) Principles of Mathematical Analysis, Third Edition, McGraw–Hill. See Theorem 7.13 on page 150 for the monotone decreasing case. Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008) [2001]. Elementary Real Analysis.
A chain in this partial order is a monotonically increasing subsequence, and an antichain is a monotonically decreasing subsequence. By Mirsky's theorem, either there is a chain of length r , or the sequence can be partitioned into at most r − 1 antichains; but in that case the largest of the antichains must form a decreasing subsequence with ...
The theorem known as the "Leibniz Test" or the alternating series test states that an alternating series will converge if the terms a n converge to 0 monotonically. Proof: Suppose the sequence a n {\displaystyle a_{n}} converges to zero and is monotone decreasing.