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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.
A function that is absolutely monotonic on [,) can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line. The big Bernshtein theorem : A function f ( x ) {\displaystyle f(x)} that is absolutely monotonic on ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} can be ...
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] More generally, the analysis of monotone functions has been studied by many mathematicians, starting from Abel, Jordan and Darboux.
A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density , the normal density , and Student's ...
A function f(x) is a weakly unimodal function if there exists a value m for which it is weakly monotonically increasing for x ≤ m and weakly monotonically decreasing for x ≥ m. In that case, the maximum value f ( m ) can be reached for a continuous range of values of x .
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 ...
A mnemonic is a memory aid used to improve long-term memory and make the process of consolidation easier. Many chemistry aspects, rules, names of compounds, sequences of elements, their reactivity, etc., can be easily and efficiently memorized with the help of mnemonics.
The function f(x) = sin(1/x) is not of bounded variation on the interval [, /]. As mentioned in the introduction, two large class of examples of BV functions are monotone functions, and absolutely continuous functions. For a negative example: the function