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The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [14] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...
The standard absolute value on the integers. The standard absolute value on the complex numbers.; The p-adic absolute value on the rational numbers.; If R is the field of rational functions over a field F and () is a fixed irreducible polynomial over F, then the following defines an absolute value on R: for () in R define | | to be , where () = () and ((), ()) = = ((), ()).
The graph of the absolute value function. If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the absolute value function = | |, [,]. Then f (−1) = f (1), but there is no c between −1 and 1 for which the f ′(c) is zero.
Hamilton's tensor is actually the absolute value on the quaternion algebra, which makes it a normed vector space. Hamilton defined tensor as a positive numerical quantity, or, more properly, signless number. [12] [13] [14] A tensor can be thought of as a positive scalar. [15] The "tensor" can be thought of as representing a "stretching factor ...
Harmonic function: its value at the center of a ball is equal to the average value on the surface of the ball (mean value property). Also subharmonic function and superharmonic function. Elementary function: composition of arithmetic operations, exponentials, logarithms, constants, and solutions of algebraic equations.
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.