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The traditional notations used in the previous section do not distinguish the original function : from the image-of-sets function : (); likewise they do not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets). Given the right context, this keeps the notation light and ...
A rectangular grid (top) and its image under a conformal map f (bottom). It is seen that f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°. A conformal map is a function which preserves angles locally. In the most common case the function has a domain and range in the complex plane. More formally, a map,
Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to prove the existence of non-measurable functions. Such proofs rely on the axiom of choice in an essential way, in the sense that Zermelo–Fraenkel set theory without the axiom of choice does not prove the existence of such functions.
The function f : R → R defined by f(x) = x 3 − 3x is surjective, because the pre-image of any real number y is the solution set of the cubic polynomial equation x 3 − 3x − y = 0, and every cubic polynomial with real coefficients has at least one real root. However, this function is not injective (and hence not bijective), since, for ...
Sometimes "range" refers to the image and sometimes to the codomain. In mathematics, the range of a function may refer to either of two closely related concepts: the codomain of the function, or; the image of the function. In some cases the codomain and the image of a function are the same set; such a function is called surjective or onto.
A function : is monotone in this topological sense if and only if it is non-increasing or non-decreasing, which is the usual meaning of "monotone function" in real analysis. A function between topological spaces is (sometimes) called a proper map if every fiber is a compact subspace of its domain. However, many authors use other non-equivalent ...
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Computational real algebraic geometry is concerned with the algorithmic aspects of real algebraic (and semialgebraic) geometry. The main algorithm is cylindrical algebraic decomposition. It is used to cut semialgebraic sets into nice pieces and to compute their projections. Real algebra is the part of algebra which is relevant to real algebraic ...