Analysis I | Part IA, 2008

(a) State and prove the intermediate value theorem.

(b) An interval is a subset II of R\mathbb{R} with the property that if xx and yy belong to II and x<z<yx<z<y then zz also belongs to II. Prove that if II is an interval and ff is a continuous function from II to R\mathbb{R} then f(I)f(I) is an interval.

(c) For each of the following three pairs (I,J)(I, J) of intervals, either exhibit a continuous function ff from II to R\mathbb{R} such that f(I)=Jf(I)=J or explain briefly why no such continuous function exists: (i) I=[0,1],J=[0,)I=[0,1], \quad J=[0, \infty); (ii) I=(0,1],J=[0,)I=(0,1], \quad J=[0, \infty); (iii) I=(0,1],J=(,)I=(0,1], \quad J=(-\infty, \infty).

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