increasing/decreasing/monotone function

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

  1. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

    f is strictly decreasing or strongly decreasing if x<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 most countableMathworldPlanetmath [1, 2].

  2. Lebesgue

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


  • 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.
Title increasing/decreasing/monotone function
Canonical name IncreasingdecreasingmonotoneFunction
Date of creation 2013-03-22 13:36:05
Last modified on 2013-03-22 13:36:05
Owner Koro (127)
Last modified by Koro (127)
Numerical id 12
Author Koro (127)
Entry type Definition
Classification msc 26A06
Classification msc 26A48
Defines increasing
Defines decreasing
Defines strictly increasing
Defines strictly decreasing
Defines monotone
Defines monotonic
Defines strictly monotone
Defines strictly monotonic
Defines weakly increasing
Defines weakly decreasing
Defines strongly increasing
Defines strongly decreasing
Defines strongly monotone
Defines weakly monotone
Defines stronly mono