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The graph of the absolute value function for real numbers The absolute value of a number may be thought of as its distance from zero. In mathematics , the absolute value or modulus of a real number x {\displaystyle x} , denoted | x | {\displaystyle |x|} , is the non-negative value of x {\displaystyle x} without regard to its sign .
Graph of the absolute value function. Note the sharp turn at x = 0, leading to non-differentiability of the curve at x = 0. The function hence possesses no ordinary derivative at x = 0. The symmetric derivative, however, exists for the function at x = 0.
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 converse, though, does not necessarily hold: for example, taking f as =, where V is a Vitali set, it is clear that f is not measurable, but its absolute value is, being a constant function. The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function.
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.
At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for y fixed and x a multiple of y the Fourier series given converges to y/2, rather than to x mod y = 0. At points of continuity the series converges to the true ...
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 ...
The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0. Piecewise defined functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.
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