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Convex Subset and Lipshitz Condition

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Please solve the following problem:

Let E be an open and convex subset of R^n and let f in C^1(E).
Show that f satisfies the Lipschitz condition on E if and only if its derivative Df is bounded on E, that is, there exists a constant M => 0 such that the norm of IIDf(x)II =< M for all x in E, where IIDf(x)II is the norm of the linear operator Df(x).

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