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单词 IncreasingdecreasingmonotoneFunction
释义

increasing/decreasing/monotone function


DefinitionLet A be a subset of , and let f be a function from f:A.Then

  1. 1.

    f is increasing or weakly increasing, ifxy implies that f(x)f(y) (for all x and y in A).

  2. 2.

    f is strictly increasing or strongly increasing, ifx<y implies that f(x)<f(y).

  3. 3.

    f is decreasing or weakly decreasing, ifxy implies that f(x)f(y).

  4. 4.

    f is strictly decreasing or strongly decreasing ifx<y implies that f(x)>f(y).

  5. 5.

    f is monotoneMathworldPlanetmath,if f is either increasing or decreasing.

  6. 6.

    f is strictly monotone or strongly monotone,if f is either strictly increasing or strictly decreasing.

Theorem Let X be a boundedPlanetmathPlanetmathPlanetmath or unboundedPlanetmathPlanetmath open interval of .In other words, let X be an interval of the form X=(a,b), where a,b{-,}.Futher, let f:X be a monotone function.

  1. 1.

    The set of points where f is discontinuousMathworldPlanetmath is at mostcountableMathworldPlanetmath [1, 2].

  2. Lebesgue

    f is differentiableMathworldPlanetmathPlanetmath almosteverywhere ([3], pp. 514).

References

  • 1 C.D. Aliprantis, O. Burkinshaw, Principles of Real Analysis,2nd ed., Academic Press, 1990.
  • 2 W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Inc., 1976.
  • 3 F. Jones, Lebesgue Integration on Euclidean Spaces,Jones and Barlett Publishers, 1993.
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