[Merged by Bors] - feat(topology/[separation, dense_embedding]): a function to a T1 space which has a limit at x is continuous at x #9934
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| @@ -125,10 +125,26 @@ lemma extend_eq_at [t2_space γ] {f : α → γ} (a : α) (hf : continuous_at f | |||
| di.extend f (i a) = f a := | |||
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Looking at this line now, it feels weird than a is an explicit argument. If you have patience, could you try to make it implicit and see whether this needs @ down the road?
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Done ! (Unless I missed some occurrence of this lemma)
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Co-authored-by: Patrick Massot <patrickmassot@free.fr>
Co-authored-by: Patrick Massot <patrickmassot@free.fr>
ADedecker
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LGTM |
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…e which has a limit at x is continuous at x (#9934) We prove that, for a function `f` with values in a T1 space, knowing that `f` admits *any* limit at `x` suffices to prove that `f` is continuous at `x`. We then use it to give a variant of `dense_inducing.extend_eq` with the same assumption required for continuity of the extension, which makes using both `extend_eq` and `continuous_extend` easier, and also brings us closer to Bourbaki who makes no explicit continuity assumption on the function to extend.
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…e which has a limit at x is continuous at x (#9934) We prove that, for a function `f` with values in a T1 space, knowing that `f` admits *any* limit at `x` suffices to prove that `f` is continuous at `x`. We then use it to give a variant of `dense_inducing.extend_eq` with the same assumption required for continuity of the extension, which makes using both `extend_eq` and `continuous_extend` easier, and also brings us closer to Bourbaki who makes no explicit continuity assumption on the function to extend.
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We prove that, for a function
fwith values in a T1 space, knowing thatfadmits any limit atxsuffices to prove thatfis continuous atx.We then use it to give a variant of
dense_inducing.extend_eqwith the same assumption required for continuity of the extension, which makes using bothextend_eqandcontinuous_extendeasier, and also brings us closer to Bourbaki who makes no explicit continuity assumption on the function to extend.