Упражнение 7
§2. Элементы общей топологии
\forall V_{y_{0}}\exists U_{x_{0}}:f(U_{x_{0}})\subset V_{y_{0}}
\forall \varepsilon > 0\: \: \exists \delta > 0\: \: \forall x: \left | x-x_{0} \right |< \delta \Rightarrow \left | f(x)-f(x_{0}) \right |< \varepsilon
\left | x-x_{0} \right |< \delta \Leftrightarrow x\in V_{y_{0}}
\left | f(x)-f(x_{0}) \right |< \varepsilon \Leftrightarrow f(x)\in B_{\varepsilon }(y_{0})\Leftrightarrow x\in f^{-1}\left ( B_{\varepsilon } (y_{0})\right )
где
y_{0}=f(x_{0})
B_{\delta }(x_{0})\subset f^{-1}\left (B_{\varepsilon }(y_{0}) \right )\Leftrightarrow f(B_{\delta }(x_{0}))\subset B_{\varepsilon }(y_{0})
\forall B_{\varepsilon }(y_{0})\: \: \exists B_{\delta }(x_{0})
B_{\varepsilon }(y_{0})\: \subset V_{y_{0}};U_{x_{0}}
\forall B_{\varepsilon }(y_{0})\: \: \exists U_{x_{0}} \left ( f(U_{x_{0}})\subset B_{\varepsilon }(y_{0}) \right )
\exists B_{\delta }(x_{0})\subset U_{x_{0}}\left ( f(B_{\delta }(x_{0}))\subset f(U_{x_{0}})\subset B_{\varepsilon }(y_{0})\right )
§2. Упражнение 7
By ASTepliakov
