Theorems · Theorem · global analysis
DifferentiableWithinAt.continuousWithinAt
∀ {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} {s : Set E} [ContinuousAdd E] [ContinuousSMul 𝕜 E] [ContinuousAdd F]
[ContinuousSMul 𝕜 F], DifferentiableWithinAt 𝕜 f s x → ContinuousWithinAt f s x- Defined in
- Mathlib.Analysis.Calculus.FDeriv.Basic
- Cited by
- 9 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.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- 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
- ContinuousWithinAtstatement and proof · cited by 512
- DifferentiableWithinAtstatement and proof · cited by 453
- HasFDerivWithinAtproof · cited by 356
Cited by9
Results whose statement or proof uses this declaration.
- DifferentiableOn.continuousOnproof · cited by 32
- mdifferentiableWithinAt_iff_differentiableWithinAtproof · cited by 5
- strictConcaveOn_of_deriv2_negproof · cited by 4
- range_derivWithin_subset_closure_span_imageproof · cited by 3
- strictMono_of_deriv_posproof · cited by 2
- strictConvexOn_of_deriv2_posproof · cited by 2
- not_differentiableWithinAt_of_deriv_tendsto_atTop_Ioiproof · cited by 2
- MeasureTheory.map_withDensity_abs_det_fderiv_eq_addHaarproof · cited by 1
- strictAnti_of_deriv_negproof · cited by 1