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Given a function f with domain D and a preordered set (K, ≤) as codomain, an element y of K is an upper bound of f if y ≥ f (x) for each x in D. The upper bound is called sharp if equality holds for at least one value of x. It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality.
A bounded operator: is not a bounded function in the sense of this page's definition (unless =), but has the weaker property of preserving boundedness; bounded sets are mapped to bounded sets (). This definition can be extended to any function f : X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle Y} allow for ...
|shift= – right-shift (padding-left) for axis origin to account for date labels (optional; default=16px) use this to vertically align the axis origin with the left margin of progress bars if labels are excluded, abbreviated, or lengthened) |ff= – fudge factor: expansion factor for tick mark placement (temp. param while in development)
If the template has a separate documentation page (usually called "Template:template name/doc"), add [[Category:Graph, chart and plot templates]] to the <includeonly> section at the bottom of that page. Otherwise, add <noinclude>[[Category:Graph, chart and plot templates]]</noinclude>
Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in a general topological space without a corresponding metric. Boundary is a distinct concept; for example, a circle (not to be confused with a disk) in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
SBV functions i.e. Special functions of Bounded Variation were introduced by Luigi Ambrosio and Ennio De Giorgi in the paper (Ambrosio & De Giorgi 1988), dealing with free discontinuity variational problems: given an open subset of , the space is a proper linear subspace of (), since the weak gradient of each function belonging to it ...
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