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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 .
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,
Trapdoor functions are a special case of one-way functions and are widely used in public-key cryptography. [2] In mathematical terms, if f is a trapdoor function, then there exists some secret information t, such that given f(x) and t, it is easy to compute x. Consider a padlock and its key. It is trivial to change the padlock from open to ...
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.
Three spin-off games accompany the main series: Geometry Dash Meltdown, Geometry Dash World and Geometry Dash SubZero. Geometry Dash Lite is a free version of the main game that removes certain levels and icons, the level editor, and many online features. Both the spinoff games and Geometry Dash Lite contain advertisements.
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map :, the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X.