Константная функция

Константная функция (также постоянная функция) — функция, которая для любого элемента из области определения возвращает одно и то же заданное значение. Заданная отображением $f\colon X\to Y$ функция $f$ является константной, если для любых элементов $x_{1},x_{2}\in X$ выполняется равенство: $f(x_{1})=f(x_{2})=y$ , где $y\in Y$ . Это означает, что множество значений функции состоит из одного единственного элемента.

Константная числовая функция $f\colon \mathbb {R} \to \mathbb {R}$ , заданная на множестве действительных чисел, является целой рациональной функцией нулевой степени или линейной функцией, график которой проходит параллельно оси абсцисс. Эта функция имеет уравнение вида $f(x)=a$ , где $a\in \mathbb {R}$ . Такая функция является чётной на всей области определения, и поэтому её график обладает осевой симметрией относительно оси ординат. Производной любой константной функции $f(x)=a$ является нулевая функция $f'(x)=0$ , которая является специальным случаем константных функций.

Преобразование Фурье константной функции $f(x)=1$ даёт дельта-функцию.

Иллюстрации

Константное отображение множеств
в декартовой системе координат на плоскости
в декартовой системе в пространстве
в полярной системе координат

См. также

Теорема Лиувилля об ограниченных целых аналитических функциях

Ссылки

MathWorld: Constant Function (англ.)

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