Monotonic function

In mathematics, ie a function or sequence that is always greater with increasing function argument or constant ( ie never falls ), monotonically increasing (or monotonically increasing or isotonic ). Accordingly, ie a function or sequence monotonically decreasing ( antitone ) when it is only smaller or remains constant. The values ​​of the function or the terms of the sequence change anywhere, it is called constant.

Strictly increasing ( resp. strictly decreasing ) are functions or sequences that are only larger (smaller), but are nowhere constant.

Monotonicity can be defined on any order relations, for example, may be monotonically increasing also refer to the subset relationship.

Examples

  • The result
  • The result
  • The result
  • The function
  • The function
  • The sequence of sets

Definitions

Let be a function. On and an order relation is defined in each case. Then the function is called monotonically increasing if for all.

Applies even so the function is called strictly monotonically increasing.

Accordingly, ie monotonically decreasing or strictly decreasing, or when.

A sequence is called monotone increasing if for all.

A sequence is called strictly monotonically increasing if for all.

Other properties

For a real monotonic function applies:

  • She has every accumulation point of its domain a left- hand and right- hand limit.
  • You can only have jumps as discontinuities.
  • The set of jump points in its domain of definition is countable, but need not necessarily be finite.
  • It is almost everywhere differentiable, ie the set of points where is not differentiable, forms a Lebesgue measure zero.
  • A predefined interval monotonic function is Riemann integrable there.

Monotony of differentiable real functions

  • A continuous on the interval and differentiable function is exactly then monotonically increasing (or monotonically decreasing ) when the derivative nowhere negative ( or positive nowhere ), so (or ) is.
  • A continuously differentiable function on an interval if and only strictly increasing (or strictly decreasing ) if the derivative: Nowhere is negative ( or anywhere positive) and
  • On any real part interval constant is equal to zero ( which is a real interval, an interval of more than one element ).

Inverse function

Be an interval and is strictly increasing / decreasing and continuous. Then:

  • The image set an interval
  • Bijective,
  • The inverse function is strictly increasing / decreasing and steadily,
  • When growing and
  • When falling.

Monotony laws

For the following applies:

Null set
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