lemma A.136 Differentiation under the integral sign

open in the book · appendices/A-long-proofs.tex:7573 · p. 2860

Rests on

depends_on definition 7.98 Functions of class $C^{1}$ ¶
- depends_on definition 7.20 Continuity at a point ¶
  - depends_on definition 7.16 Limit ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on equation 7.7 eq:ana-limit-left ¶
  - depends_on equation 7.5 eq:ana-limit-right ¶
- depends_on definition 7.97 Partial derivative; gradient ¶
  - depends_on definition 7.26 Derivative of a function at a point ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on definition 5.15 Basis ¶
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
depends_on theorem 6.11 Heine–Borel on $\R$ ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on equation 3.51 eq:set-indexed ¶
- proves proof ch:04-topology@proof-2 ¶
proves proof app:A-long-proofs@proof-83 ¶

Supports

depends_on proposition A.142 The mollification is smooth ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Functions of class $C^{1}$	declared	appendices/A-long-proofs.tex:7587
`depends_on`	→	Heine–Borel on $\R$	declared	appendices/A-long-proofs.tex:7587
`depends_on`	←	The mollification is smooth	declared	appendices/A-long-proofs.tex:7794
`proves`	←	app:A-long-proofs@proof-83	declared	appendices/A-long-proofs.tex:7590