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ε-δ_function_limit [2014/03/11 09:31]
nikolaj
ε-δ_function_limit [2015/01/15 10:09]
nikolaj
Line 1: Line 1:
 ===== ε-δ function limit ===== ===== ε-δ function limit =====
 ==== Set ==== ==== Set ====
-| @#88DDEE$\langle X,​d_X\rangle$ ... metric space | +>todo:  
-| @#88DDEE: $\langle Y,​d_Y\rangle$ ... metric space | +>Rather define the set of all sequences which have a limit in the below sense 
-| @#88DDEE: $f:X\to Y$ |  +>this is the domain of "​lim"​ though of as function (on all sequences it would be a partial function)
-| @#88DDEE: $\xi\in X$ |+
  
-| @#FFBB00: $\mathrm{lim}_{x\to \xi}\ f(x)\equiv y_\xi$ |+| @#55CCEE: context ​    | @#55CCEE: $\langle X,d_X\rangle$ ... metric space | 
 +| @#55CCEE: context ​    | @#55CCEE: $\langle Y,d_Y\rangle$ ... metric space | 
 +| @#55CCEE: context ​    | @#55CCEE: $f:X\to Y$ |  
 +| @#55CCEE: context ​    | @#55CCEE: $\xi\in X$ |
  
-| @#DDDDDD: $\varepsilon,\delta\in \mathbb R_+^*$ |+| @#FFBB00: definiendum | @#FFBB00: $\mathrm{lim}_{x\to \xi}f(x)\equiv y_\xi$ |
  
-| @#55EE55: $\forall\varepsilon.\ \exists \delta.\ \forall x.\ [\ 0<​d_X(x,​\xi)<​\delta\ ] \Rightarrow [\ d_Y(f(x),​y_\xi)<​\varepsilon\ ]$ | +| @#DDDDDD: range       | @#DDDDDD: $\varepsilon,​\delta\in \mathbb R_+^*$ | 
 + 
 +| @#55EE55: postulate ​  | @#55EE55: $\forall\varepsilon.\ \exists \delta.\ \forall x.\ [\ 0<​d_X(x,​\xi)<​\delta\ ] \Rightarrow [\ d_Y(f(x),​y_\xi)<​\varepsilon\ ]$ | 
  
 ==== Discussion ==== ==== Discussion ====
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