Monotonic functions

Japanese: 単調関数 - たんちょうかんすう

A term that refers to both increasing and decreasing functions. Since a decreasing function becomes an increasing function when you add a minus sign, the following explanation will focus on increasing functions. For a real-valued function f(x) defined in a certain interval a≦x≦b (one or both ends of the interval do not have to be included, and a and b can be ±∞),
If _x1 < _x2 , then f( _x1 ) ≤ f( _x2 )
If f(x 1 ) < f(x 2 ), then f(x) is said to be monotonically increasing in this interval. If f(x ₁ ) < f(x ₂ ) is always true, then it is simply called an increasing (or strictly increasing) function. For example,
f(x)=x (-∞<x<+∞)
f(x)= ^x2 (x≧0)
etc. are increasing functions. Also,
f(x)=［x］ (-∞＜x＜+∞)
is also a monotonically increasing function ( Figure A ). Here, [x] represents the integer part of x, i.e., the largest integer not greater than x. [x] is called the Gauss symbol. A function in which the entire range is divided into several intervals like this and increases while taking on a constant value in each small interval is called a step function.

A monotonically increasing function has the following properties:

(1) A monotonically increasing function is not necessarily continuous, but there are at most countable number of discontinuous points, and each discontinuous point has a limit value on the left and a limit value on the right ( Figure B ).

(2) A continuous, increasing function has an inverse function that is also continuous and increasing.

(3) A monotonically increasing function is differentiable at almost all points, i.e., except on the set whose Lebesgue measure is zero.

(4) If f(x) is differentiable at all points and f′(x)≧0, then f(x) is monotonically increasing.

(5) A monotonically increasing function is Riemann integrable.

(6) A monotonically increasing function f(x) on the interval [a, b] can be expanded into a Fourier series on [a, b]. That is, the Fourier series converges at all points and has the same value as f(x) at any continuous point of f(x). This was the first step in the theory of Fourier series made by Dirichlet (1829).

A function that can be expressed as the difference of monotonically increasing functions is called a bounded variation function. It is characterized by the following property. If c and d (c < d) are any two points in the domain, then there is a positive number M, and
c=x ₀ ＜x ₁ ＜x ₂ ＜...＜x _n =d
For any point x ₁ , x ₂ , …, xn _-1, |f(x ₁ )-f(x ₀ )|
+|f( _x2 )-f( _x1 )|
+……+|f(x _n )-f(x _n-1 )|≦M
It becomes.

[Osamu Takenouchi]

[Reference] | Fourier series | Riemann integral

Monotonically increasing function (Figure A)
©Shogakukan ">

Monotonically increasing function (Figure A)

Properties of monotonically increasing functions (Figure B)
©Shogakukan ">

Properties of monotonically increasing functions (Figure B)

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

増加関数、減少関数をあわせていう術語。減少関数は、マイナスをつければ増加関数になるから、以下の説明では増加関数について述べる。実数のある区間a≦x≦b（区間の片方、または両方の端が入っていなくてもよい。またa、bは±∞でもよい）において定義された実数値関数f(x)に対し、
　　x₁＜x₂　ならば　f(x₁)≦f(x₂)
であるとき、f(x)はこの区間で単調増加であるという。もし、ここで、つねにf(x₁)＜f(x₂)であるときは、単に増加（または狭義の増加）関数という。たとえば、
　　f(x)=x　(-∞＜x＜+∞)
　　f(x)=x²　(x≧0)
などは増加関数である。また、
　　f(x)=［x］　(-∞＜x＜+∞)
も単調増加関数である（図A）。ここで［x］はxの整数部分、すなわちxより大きくない整数のうちで最大のものを表す。［x］はガウスの記号とよばれている。このように全区間がいくつかの区間に分けられ、各小区間の上では定数値をとりながら増加していくような関数を階段関数という。