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ε-δ_function_limit [2014/02/13 16:13] 127.0.0.1 external edit |
ε-δ_function_limit [2015/01/15 10:09] nikolaj |
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| ===== 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 ==== | ||
| - | === Reference === | ||
| - | Wikipedia: [[https://en.wikipedia.org/wiki/%28%CE%B5,_%CE%B4%29-definition_of_limit|Epsilon-delta definition of a limit]] | ||
| ==== Parents ==== | ==== Parents ==== | ||
| - | === Related === | + | === Context === |
| [[Metric space]] | [[Metric space]] | ||
