Search results
Results from the WOW.Com Content Network
Example 2: a function f is uniformly continuous on the semi-open interval [0,∞) if and only if it is continuous at the standard points of the interval, and in addition, the natural extension f* is microcontinuous at every positive infinite hyperreal point. Example 3: similarly, the failure of uniform continuity for the squaring function
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]
respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. [7] If the one-sided limits exist at p, but are unequal, then there is no limit at p (i.e., the limit at p does not exist). If either one-sided limit does not exist at p, then the limit at p also does not exist.
The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.
An animated example of a Brownian motion-like random walk on a torus.In the scaling limit, random walk approaches the Wiener process according to Donsker's theorem.. In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model characterizes its behaviour in the limit as the lattice spacing goes to zero.
A functor G : C → D is said to lift limits for a diagram F : J → C if whenever (L, φ) is a limit of GF there exists a limit (L′, φ′) of F such that G(L′, φ′) = (L, φ). A functor G lifts limits of shape J if it lifts limits for all diagrams of shape J. One can therefore talk about lifting products, equalizers, pullbacks, etc.
In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form , = (,), (,) = ((,)),or other similar forms. An iterated limit is only defined for an expression whose value depends on at least two variables. To evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value ...
The value of this limit, should it exist, is the (C, α) sum of the integral. An integral is (C, 0) summable precisely when it exists as an improper integral. However, there are integrals which are (C, α) summable for α > 0 which fail to converge as improper integrals (in the sense of Riemann or Lebesgue).