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Консультация № 191353
16.09.2017, 21:01
0.00 руб.
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Здравствуйте! Прошу помощи в следующем вопросе:
Прикрепленные файлы:
Обсуждение
17.09.2017, 14:34
общий
Адресаты:
Пусть 
- функция, непрерывная на полуотрезке 
и 
при 
Нужно доказать, что функция ограничена на полуотрезке 
"Очевидно", что можно взять функцию
непрерывную на отрезке 
Эта функция, согласно теореме Вейерштрасса, ограничена на отрезке "Выкалыванием" точки 
получим заданную функцию 
которая остаётся ограниченной.
Наверное, остаётся доказать это строго, со всеми необходимыми ссылками на определения и теоремы...
- функция, непрерывная на полуотрезке и при Нужно доказать, что функция ограничена на полуотрезке "Очевидно", что можно взять функцию
непрерывную на отрезке Эта функция, согласно теореме Вейерштрасса, ограничена на отрезке "Выкалыванием" точки получим заданную функцию которая остаётся ограниченной.Наверное, остаётся доказать это строго, со всеми необходимыми ссылками на определения и теоремы...
Об авторе:
Facta loquuntur.
Facta loquuntur.
17.09.2017, 14:58
общий
Адресаты:
Как мне подсказали на другом форуме, указанного выше рассуждения хватает для доказательства. Функция 
получена из функции доопределением по непрерывности в точке 
Функция совпадает с функцией на полуотрезке 
и поэтому ограничена на нём.
получена из функции доопределением по непрерывности в точке Функция совпадает с функцией на полуотрезке и поэтому ограничена на нём.
Об авторе:
Facta loquuntur.
Facta loquuntur.
17.09.2017, 15:09
общий
это ответ
Здравствуйте, Елена Васильевна!
Рассмотрим задание в общей формулировке.
Пусть - функция, непрерывная на полуотрезке и при Нужно доказать, что функция ограничена на полуотрезке
Возьмём функцию непрерывную на отрезке 
доопределив по непрерывности в точке 
то есть приняв 
Эта функция, согласно теореме Вейерштрасса, ограничена на отрезке Функция совпадает с функцией на полуотрезке и поэтому ограничена на нём.
Рассмотрим задание в общей формулировке.
Пусть
- функция, непрерывная на полуотрезке и при Нужно доказать, что функция ограничена на полуотрезке Возьмём функцию
непрерывную на отрезке доопределив по непрерывности в точке то есть приняв Эта функция, согласно теореме Вейерштрасса, ограничена на отрезке Функция совпадает с функцией на полуотрезке и поэтому ограничена на нём.
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Об авторе:
Facta loquuntur.
Facta loquuntur.
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