Матеріал з Вікіпедії — вільної енциклопедії.
Подстраница "Користувач:Галактион/Друга теорема Вейєрштрасса" создана для того, чтобы перенести информацию из раздела "Обговорення" статьи "Друга теорема Вейєрштрасса ". Галактион 03:51, 5 березня 2010 (UTC)
Вторая теорема Вейерштрасса формулируется на языке сообщества математиков так:
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{\displaystyle {\begin{aligned}t\vdash \quad \mathbb {X} \times \mathbb {Y} \subseteq \mathbb {R} ^{2}\quad \land \quad \mathrm {f} :\mathbb {X} \mapsto \mathbb {Y} \quad \land \quad \{a,b\}\subseteq \mathbb {R} \quad \land \quad a<b\quad \land \quad [a,b]\subseteq \mathbb {X} \\\ \land \quad \mathrm {f} \in \mathrm {C_{ont}} ([a,b])\quad \to \quad \exists _{\{c,d\}\ \subseteq \ [a,b]}\ \forall _{x\ \in \ [a,b]}\ (f(c)\ \leq \ f(x)\ \leq \ f(d))\end{aligned}}}
Примечание
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{\displaystyle ~\mathrm {f} :\mathbb {X} \mapsto \mathbb {Y} \quad \Leftrightarrow \quad \mathrm {f} =\{\langle x,y\rangle |\ \ \langle x,y\rangle \in \mathbb {X} \times \mathbb {Y} \ \land \ y=f(x)\}}
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{\displaystyle ~\{a,b\}\subseteq \mathbb {R} \quad \Leftrightarrow \quad a\in \mathbb {R} \ \land \ b\in \mathbb {R} }
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{\displaystyle ~[a,b]\subseteq \mathbb {R} \quad \Leftrightarrow \quad \{x|\ \ x\in \mathbb {R} \ \land \ a\leq x\leq b\}\subseteq \mathbb {R} }
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{\displaystyle ~\vdash \quad (a,b)=\{x|\ \ x\in \mathbb {R} \ \land \ a<x<b\}}
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{\displaystyle ~\mathrm {f} \in \mathrm {C_{ont}} ([a,b])\quad \Leftrightarrow \quad f(a)=\lim _{t\to a^{+}}f(t)\quad \land \quad \forall _{x\ \in \ (a,b)}\ (f(x)=\lim _{t\to x}f(t))\quad \land \quad f(b)=\lim _{t\to b^{-}}f(t)}
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{\displaystyle ~\exists _{\{c,d\}\ \subseteq \ [a,b]}\ \forall _{x\ \in \ [a,b]}\ (f(c)\leq f(x)\leq f(d))\ \Leftrightarrow }
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{\displaystyle ~\exists c\exists d\ (c\in [a,b]\ \land \ d\in [a,b]\ \ \land \ \ \forall x\ (x\in [a,b]\ \to \ f(c)\ \leq \ f(x)\ \ \land \ \ f(x)\ \leq \ f(d))\ )}
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{\displaystyle ~\exists _{\{c,d\}\ \subseteq \ [a,b]}\ \forall _{x\ \in \ [a,b]}\ (f(c)\leq f(x)\leq f(d)\quad \Leftrightarrow }
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{\displaystyle ~\exists _{c\ \in \ [a,b]}\ \forall _{x\ \in \ [a,b]}\ (f(x)=f(c))\quad \lor \quad \exists _{\{c,d\}\ \subseteq \ [a,b]\ \land \ c\ \neq \ d}\ \forall _{x\ \in \ [a,b]}\ (f(c)\leq f(x)\leq f(d))}
Галактион 18:11, 26 серпня 2009 (UTC)
![{\displaystyle {\begin{aligned}t\vdash \quad \mathbb {X} \times \mathbb {Y} \subseteq \mathbb {R} ^{2}\quad \land \quad \mathrm {f} :\mathbb {X} \mapsto \mathbb {Y} \quad \land \quad \{a,b\}\subseteq \mathbb {R} \quad \land \quad a<b\quad \land \quad [a,b]\subseteq \mathbb {X} \\\ \land \quad \mathrm {f} \in \mathrm {C_{ont}} ([a,b])\quad \to \quad \exists _{\{c,d\}\ \subseteq \ [a,b]}\ \forall _{x\ \in \ [a,b]}\ (f(c)\ \leq \ f(x)\ \leq \ f(d))\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a81597762463706fa14728cc164d5163c247c80)
![{\displaystyle ~[a,b]\subseteq \mathbb {R} \quad \Leftrightarrow \quad \{x|\ \ x\in \mathbb {R} \ \land \ a\leq x\leq b\}\subseteq \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f47b2ced24752ca2bff8b2ccbdf9466a9c2147e7)
![{\displaystyle ~\mathrm {f} \in \mathrm {C_{ont}} ([a,b])\quad \Leftrightarrow \quad f(a)=\lim _{t\to a^{+}}f(t)\quad \land \quad \forall _{x\ \in \ (a,b)}\ (f(x)=\lim _{t\to x}f(t))\quad \land \quad f(b)=\lim _{t\to b^{-}}f(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c85abe27d21115665f99fdfd687d7095469d3181)
![{\displaystyle ~\exists _{\{c,d\}\ \subseteq \ [a,b]}\ \forall _{x\ \in \ [a,b]}\ (f(c)\leq f(x)\leq f(d))\ \Leftrightarrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/185f4e870c7b88d070d479b21904f5f1a2761749)
![{\displaystyle ~\exists c\exists d\ (c\in [a,b]\ \land \ d\in [a,b]\ \ \land \ \ \forall x\ (x\in [a,b]\ \to \ f(c)\ \leq \ f(x)\ \ \land \ \ f(x)\ \leq \ f(d))\ )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e71587394334190dfea6a62df41f6bf89b920e66)
![{\displaystyle ~\exists _{\{c,d\}\ \subseteq \ [a,b]}\ \forall _{x\ \in \ [a,b]}\ (f(c)\leq f(x)\leq f(d)\quad \Leftrightarrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/43102995a13632daa31eb992f4bff54deecdc330)
![{\displaystyle ~\exists _{c\ \in \ [a,b]}\ \forall _{x\ \in \ [a,b]}\ (f(x)=f(c))\quad \lor \quad \exists _{\{c,d\}\ \subseteq \ [a,b]\ \land \ c\ \neq \ d}\ \forall _{x\ \in \ [a,b]}\ (f(c)\leq f(x)\leq f(d))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54fe51fcb2c0a9ce8070d9f80e9f13a6c89210ff)