# Positive Number Assignment | Assignment Help Services

1)  Suppose A ⊆ Rd. A function f : A → Rm is said to be Lipschitz if there is a positive number M such that ||f(x) − f(y)|| ≤ M||x − y|| ∀ x, y ∈ A. (a) Show that a Lipschitz function must be uniformly continuous.(b) Show that the function h: [0, 1] → R, h(x) = √x,is uniformly continuous on [0, 1], but is not Lipschitz.

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2) Suppose K is a compact subset of Rd and, for all k1, k2 ∈ K, there exists a continuous function p: [0, 1] → K such that p(0) = k1 and p(1) = k2. (A set with this property is said to be path connected.)Let f : K → R be continuous on K. Prove that there exist kmin and kmax such that f(K) = [f(kmin), f(kmax)].Hint: If p: [0, 1] → K and f : K → R are continuous, then the composition f ◦ p: [0, 1] → R is continuous.

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