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Problem 21
21. Given two open real intervals U,V and a function f: U -> V which is strictly monotonous and onto, show that f and f^(-1) are continuous.
Solution

Problem 22
22. Given an upper semi-continuous function f on [0,1], show that there exists a sequence of decreasing continuous functions (f_n) converging to f.
Solution

Problem 23
23. Show that [0,1] cannot be written as the countably infinite union of closed subintervals.
Solution

Problem 24
24. Give a proof of the Intermediate Values Theorem using the fact that, given a continuous real function f: R -> R, for every f(b) > gamma > f(a), f(sup({x in [a,b]: f(x) <= gamma })=gamma.
Solution



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