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The graph of the absolute value function for real numbers Composition of absolute value with a cubic function in different orders. The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It is monotonically decreasing on the interval (−∞, 0] and monotonically increasing on the interval [0 ...
A subderivative value 0 occurs here because the absolute value function is at a minimum. The full family of valid subderivatives at zero constitutes the subdifferential interval [ − 1 , 1 ] {\displaystyle [-1,1]} , which might be thought of informally as "filling in" the graph of the sign function with a vertical line through the origin ...
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x 1/n).
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 ((), ()) = = ((), ()).
Constant function: polynomial of degree zero, graph is a horizontal straight line; Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola. Cubic function: Third degree polynomial. Quartic function: Fourth degree polynomial. Quintic function: Fifth degree polynomial.
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.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
The absolute value of a complex number is a positively homogeneous function of degree over the real numbers (that is, when considering the complex numbers as a vector space over the real numbers). It is not homogeneous, over the real numbers as well as over the complex numbers.
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