Search results
Results from the WOW.Com Content Network
A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x 0 is an interior point in the domain of a function f, then f is said to be differentiable at x 0 if the derivative ′ exists.
If there exists an m × n matrix A such that = + ‖ ‖ in which the vector ε → 0 as Δx → 0, then f is by definition differentiable at the point x. The matrix A is sometimes known as the Jacobian matrix , and the linear transformation that associates to the increment Δ x ∈ R n the vector A Δ x ∈ R m is, in this general setting ...
A function of a real variable is differentiable at a point of its domain, if its domain contains an open interval containing , and the limit = (+) exists. [2] This means that, for every positive real number , there exists a positive real number such that, for every such that | | < and then (+) is defined, and | (+) | <, where the vertical bars denote the absolute value.
The C 0 function f (x) = x for x ≥ 0 and 0 otherwise. The function g (x) = x 2 sin(1/ x) for x > 0. The function : with () = for and () = is differentiable. However, this function is not continuously differentiable.
The function f(x) defined in the paragraph above is a counterexample, as it is not defined for x = ±i. This explains why the Taylor series of f ( x ) diverges for | x | > 1, i.e., the radius of convergence is 1 because the complexified function has a pole at distance 1 from the evaluation point 0 and no further poles within the open disc of ...
the function f is n − 1 times continuously differentiable on the closed interval [a, b] and the n th derivative exists on the open interval (a, b), and; there are n intervals given by a 1 < b 1 ≤ a 2 < b 2 ≤ ⋯ ≤ a n < b n in [a, b] such that f (a k) = f (b k) for every k from 1 to n. Then there is a number c in (a, b) such that the n ...
Furthermore, g(x) = 0 for x ≤ 0 and g(x) = 1 for x ≥ 1, hence it provides a smooth transition from the level 0 to the level 1 in the unit interval [0, 1]. To have the smooth transition in the real interval [a, b] with a < b, consider the function
Thus, formally, with x = (x 1, x 2, ..., x n), one has [26] ... If Y = g(X) is a continuous differentiable function, then the density of Y can be written as = ...