feat(analysis/calculus/extend_deriv): extend differentiability to the boundary #1794
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jcommelin
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PatrickMassot
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I don't really like that the main proof uses sequences in our pedantic abstract topology library. More seriously, it looks really long. But i'm not sure the sequence-free proof will be really shorter (or more readable). So let's merge that and then I'll try another proof, and PR it if it looks good.
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… boundary (leanprover-community#1794) * feat(analysis/calculus/extend_deriv): extend differentiability to the boundary * fix build Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>
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… boundary (leanprover-community#1794) * feat(analysis/calculus/extend_deriv): extend differentiability to the boundary * fix build Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>
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If a function is differentiable inside a convex open set, and the function and its derivative admit a limit on the boundary, then the function is also differentiable at the boundary point. We give a general version of this statement, and specialize it to one-dimensional situations.
Motivation: in a later PR, I will show that
x -> exp(-1/x)forx > 0and0otherwise is smooth, aiming at smooth partitions of unity. The only difficulty is to discuss its differentiability at0, which is done thanks to the statements we prove here.