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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.
Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not. In mathematics , a function f {\displaystyle f} defined on some set X {\displaystyle X} with real or complex values is called bounded if the set of its values is bounded .
|square= Makes the chart/plot a square (default no) |width= The width of the chart |picture= The picture for the background of the chart, excluding File: or Image: (default Blank.png) |size= The size of the dots (default 8px) |bottom= Text tho show on the bottom of the template |top= The header to show on top of the graph
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.
This graph draws one or more independent numeric data series as lines. The data must be stored on Commons' Data namespace or come from Wikidata Query Service. Template parameters Parameter Description Type Status Table type tabletype Specifies the type of the table data. "tab" (default) uses data namespace on commons, without the data: prefix. "query" sends request to wikidata query service ...
Let : a function between topological vector spaces is said to be a locally bounded function if every point of has a neighborhood whose image under is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces:
A logarithmic chart allows only positive values to be plotted. A square root scale chart cannot show negative values. x: the x-values as a comma-separated list, for dates and time see remark in xType and yType; y or y1, y2, …: the y-values for one or several data series, respectively. For pie charts y2 denotes the radius of the corresponding ...
In graph theory, a -bounded family of graphs is one for which there is some function such that, for every integer the graphs in with = (clique number) can be colored with at most () colors. The function f ( t ) {\displaystyle f(t)} is called a χ {\displaystyle \chi } -binding function for F {\displaystyle {\mathcal {F}}} .