6. Suppose that E ~ Rn and I : E ~ Rm.

(a) Prove that I is continuous on E if and only if I-I (V) is relatively open in E for every open set V in R m.

(b) Prove that I is continuous on E if and only if 1-1 (B) is relatively closed in E for every closed set B in R m.

(c) Suppose that I is continuous on E. Prove that if V is relatively open in I(E), then 1-1 (V) is relatively open in E, and if B is relatively closed in I(E), then 1-1 (B) is relatively closed in E.

trJ. This exercise is used in Section e9.5.

Let n,m E N and let H be a nonempty, closed, bounded subset of Rn.

(a) Suppose that I : H ~ Rm is continuous. Prove that IIIIIH := sup 111(x)11

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