Theorems · Theorem · global analysis
DifferentiableAt.continuousAt
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
[inst_3 : TopologicalSpace E] {F : Type u_3} [inst_4 : AddCommGroup F] [inst_5 : Module 𝕜 F]
[inst_6 : TopologicalSpace F] {f : E → F} {x : E} [ContinuousAdd E] [ContinuousSMul 𝕜 E] [ContinuousAdd F]
[ContinuousSMul 𝕜 F], DifferentiableAt 𝕜 f x → ContinuousAt f x- Defined in
- Mathlib.Analysis.Calculus.FDeriv.Basic
- Cited by
- 28 results in Mathlib
- Foundations
- Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- RingHom.idproof · cited by 18,349
- AddCommGroupstatement and proof · cited by 12,871
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- ContinuousLinearMapproof · cited by 5,352
- ContinuousSMulstatement and proof · cited by 1,016
- ContinuousAddstatement and proof · cited by 777
- ContinuousAtstatement and proof · cited by 697
- DifferentiableAtstatement and proof · cited by 617
- HasFDerivAtproof · cited by 350
- HasFDerivAt.continuousAtproof · cited by 8
Cited by28
Results whose statement or proof uses this declaration.
- Differentiable.continuousproof · cited by 28
- differentiableAt_zpowproof · cited by 5
- Complex.differentiableOn_compl_singleton_and_continuousAt_iffproof · cited by 4
- mdifferentiableAt_iff_differentiableAtproof · cited by 4
- Real.convexOn_Gammaproof · cited by 2
- Real.deriv_log_log_applyproof · cited by 2
- Real.deriv2_mul_logproof · cited by 2
- HurwitzZeta.tendsto_hurwitzZetaEven_sub_one_div_nhds_oneproof · cited by 2
- Real.deriv2_qaryEntropyproof · cited by 2
- Real.deriv_inv_log_applyproof · cited by 2
- AnalyticOnNhd.hasFTaylorSeriesUpToOnproof · cited by 1