QUESTION 5 Consider the following statements: (i) There exists a function f: R→ R that is discontinuous at every rational number and continuous at every irrational number. (ii) There exists a function f:R→R that is continuous and such that f(R) = Q. Which statements are true? Only (ii) is true. O Only (i) is true. Both are true. Neither are true.

(1) For each of the following definitions, (a) rewrite the definition using math symbols wherever possible.; (b) write the negation of the definition using math symbols wherever possible; (c) write the negation of the definition in English. (1.1) A function f: R → R is unbounded if and only if given any M E R, there exists an x E R such that the absolute value of f (x) is greater than or equal to M. (1.2) A subset K of R is compact if and only if for any collection of open subsets of {U;}, whose union contains the set K, there exists a finite sub-collection {Uj, Uja,' Uin} of {U;}1 such ... that K is contained in the union of this finite sub-collection of sets.

4 Q Search 2 C 8. Suppose that the function f : [0,1] → R is continuous and its image consist en- tirely of rational numbers. Prove that f is constant.

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