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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 ...
A function is unimodal if it is monotonically increasing up to some point (the mode) and then monotonically decreasing. When f {\displaystyle f} is a strictly monotonic function, then f {\displaystyle f} is injective on its domain, and if T {\displaystyle T} is the range of f {\displaystyle f} , then there is an inverse function on T ...
There is also a connection formula for the ratio of two rising factorials given by () = (+) (),. Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities: [11] (p 52)
If is a compact topological space, and () is a monotonically increasing sequence (meaning () + for all and ) of continuous real-valued functions on which converges pointwise to a continuous function :, then the convergence is uniform.
A benefit of isotonic regression is that it is not constrained by any functional form, such as the linearity imposed by linear regression, as long as the function is monotonic increasing. Another application is nonmetric multidimensional scaling , [ 1 ] where a low-dimensional embedding for data points is sought such that order of distances ...
The first-derivative test depends on the "increasing–decreasing test", which is itself ultimately a consequence of the mean value theorem. It is a direct consequence of the way the derivative is defined and its connection to decrease and increase of a function locally, combined with the previous section.
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]
Let (f n) n ∈ N be a sequence of increasing functions mapping a real interval I into the real line R, and suppose that it is uniformly bounded: there are a,b ∈ R such that a ≤ f n ≤ b for every n ∈ N. Then the sequence (f n) n ∈ N admits a pointwise convergent subsequence.