A schematic illustration of a bounded function (red) and an unbounded one (blue). 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 M such that
- X. A function that is not bounded is said to be unbounded.f(x)≤A for all A. If X, then the function is said to be bounded (from) below by X is taken to be the set M such that
- l∞.Y by requiring that the image
T(M)⊆Y. This definition can be extended to any function Y allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.|sin(x)|≤1 for all x except for −1 and 1, is unbounded. As (−∞,−2].x, is bounded, since
- The inverse trigonometric function arctangent defined as: x and bounded with
- All complex-valued functions
- The function x irrational number (cf. Dirichlet function) is bounded. Thus, a function does not need to be "nice" in order to be bounded. The set of all bounded functions defined on g:ℝ2→ℝ and h(x,y):=1x+y are both continuous, but neither is bounded. (However, a continuous function must be bounded if its domain is both closed and bounded.)
See also
editReferences
editPolyanin, Andrei D.; Chernoutsan, Alexei (2010-10-18). A Concise Handbook of Mathematics, Physics, and Engineering Sciences. CRC Press. ISBN 978-1-4398-0640-1.- ↑ Ghorpade, Sudhir R.; Limaye, Balmohan V. (2010-03-20). A Course in Multivariable Calculus and Analysis. Springer Science & Business Media. p. 56. ISBN 978-1-4419-1621-1.
- l∞.Y by requiring that the image
T(M)⊆Y. This definition can be extended to any function Y allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.|sin(x)|≤1 for all x except for −1 and 1, is unbounded. As (−∞,−2].x, is bounded, since
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