Paper 1, Section II, F
Let be a non-empty open set and let .
(a) What does it mean to say that is differentiable? What does it mean to say that is a function?
If is differentiable, show that is continuous.
State the inverse function theorem.
(b) Suppose that is convex, is and that its derivative at a satisfies for all , where is the identity map and denotes the operator norm. Show that is injective.
Explain why is an open subset of .
Must it be true that ? What if ? Give proofs or counter-examples as appropriate.
(c) Find the largest set such that the map given by satisfies for every .
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