Detachment and Tendency of a Single Variable Function

Depicted are the pointwise operators: "detachment from left or right" and "tendency" applied to a given function. In the suggested graphs, the value of the one-sided detachments is colored in green, and the value of the tendency is colored in red.
The definition of the detachment is intuitive: the detachment from the left or right is the limit (from the left or right) of the sign of . The definition of the tendency is based on the one-sided detachments: if they are equal, then the tendency is 0; otherwise, the tendency equals the right detachment. Thus, a function's tendency in many cases simulates the sign of the derivative—without calculating the derivative first (this calculation saving has advantages in theory and applications).

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Ever since the early 1980s, computer scientists have been using discrete versions of Green's and Stokes's theorems, also known as the "summed area tables" algorithms. These theorems were shown to provide a tremendous computational gain, since they fit precisely to the needs of discrete geometry researchers due to their discrete nature. It was suggested that these theorems are actually derived from a differently defined calculus, namely the "calculus of detachment". The main operator of this theory is defined by a mixture of the discrete and continuous to form a semi-discrete version of the familiar derivative. This approach to analyze functions is hence more suitable for computers (in order to save computation time), and the simplicity of the definition allows further research in other areas of classical analysis. A basic and not very comprehensive video introduction to this theory is available, and a preprint is also available.


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