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In physics, for example, the space-time continuum model describes space and time as part of the same continuum rather than as separate entities. A spectrum in physics, such as the electromagnetic spectrum, is often termed as either continuous (with energy at all wavelengths) or discrete (energy at only certain wavelengths).
The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687, [2] which was the first problem in the field to be formulated and correctly solved, [2] and was also one of the most difficult problems tackled by variational methods prior to the twentieth century.
Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions. Another characterization states that the functions of bounded variation on a compact interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone ...
If f is a Lipschitz continuous -valued one-form on , and X is a continuous function from the interval [a, b] to with finite p-variation with p less than 2, then the integral of f on X, (()) (), can be calculated because each component of f(X(t)) will be a path of finite p-variation and the integral is a sum of finitely many Young integrals.
f: I → R is absolutely continuous if and only if it is continuous, is of bounded variation and has the Luzin N property. This statement is also known as the Banach-Zareckiǐ theorem. [8] If f: I → R is absolutely continuous and g: R → R is globally Lipschitz-continuous, then the composition g ∘ f is absolutely continuous.
A set A of functions continuous between two topological spaces X and Y is topologically equicontinuous at the points x ∈ X and y ∈ Y if for any open set O about y, there are neighborhoods U of x and V of y such that for every f ∈ A, if the intersection of f[U] and V is nonempty, f[U] ⊆ O.
The given functions (f, g) may be discontinuous, provided that they are locally integrable (on the given interval). In this case, Lebesgue integration is meant, the conclusions hold almost everywhere (thus, in all continuity points), and differentiability of g is interpreted as local absolute continuity (rather than continuous differentiability).
An adapted continuous process is a quadratic pure-jump semimartingale if and only if it is of finite variation. For every semimartingale X there is a unique continuous local martingale X c {\displaystyle X^{c}} starting at zero such that X − X c {\displaystyle X-X^{c}} is a quadratic pure-jump semimartingale ( He, Wang & Yan 1992 , p. 209 ...