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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,
In general, any measurable function can be pushed forward. The push-forward then becomes a linear operator, known as the transfer operator or Frobenius–Perron operator.In finite spaces this operator typically satisfies the requirements of the Frobenius–Perron theorem, and the maximal eigenvalue of the operator corresponds to the invariant measure.
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.
Shape analysis is the (mostly) [clarification needed] automatic analysis of geometric shapes, for example using a computer to detect similarly shaped objects in a database or parts that fit together. For a computer to automatically analyze and process geometric shapes, the objects have to be represented in a digital form.
Geometry Dash is a side-scrolling music platforming game series developed by Robert Topala. It was released on 13 August 2013 for iOS and Android, with versions for Windows and macOS following on 22 December 2014. In Geometry Dash, players control an icon to navigate music-based levels, avoiding obstacles like spikes.