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The image of the function is the set of all output values it may produce, that is, the image of . The preimage of f {\displaystyle f} , that is, the preimage of Y {\displaystyle Y} under f {\displaystyle f} , always equals X {\displaystyle X} (the domain of f {\displaystyle f} ); therefore, the former notion is rarely used.
Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see Luzin's theorem . If a Borel function happens to be a section of a map Y → π X , {\displaystyle Y\xrightarrow {~\pi ~} X,} it is called a Borel section .
is a function from domain X to codomain Y. The yellow oval inside Y is the image of . 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.
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 ...
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,
In other words, every element of the function's codomain is the image of at most one element of its domain. [2] The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.
If dealing with sheaves of sets instead of sheaves of abelian groups, the same definition applies. Similarly, if f: (X, O X) → (Y, O Y) is a morphism of ringed spaces, we obtain a direct image functor f ∗: Sh(X,O X) → Sh(Y,O Y) from the category of sheaves of O X-modules to the category of sheaves of O Y-modules.
The formula is a general-purpose method of decoding a bitmap stored in the constant , and it could be used to draw any other image. When applied to the unbounded positive range 0 ≤ y {\displaystyle 0\leq y} , the formula tiles a vertical swath of the plane with a pattern that contains all possible 17-pixel-tall bitmaps.