(5) (15 points) Let f : Rn → Rm be continuous on Rn and let B be a bounded subset in Rn. Prove or disprove (if you think the following statement is false, give a counter-example and prove that your example works) that f(B) is bounded.

(4) (20 points) Every rational t can be written in the form, where n> 0, and m and n are integers without any common divisors. When a = O, we take n = 1. Consider the function f defined on R by Prove that f is continuous at every irrational point, and that f has a discontinuity at every rational point.

(5) (15 points) Let f : Rⁿ → R^m be continuous on Rⁿ and let B be a bounded subset in Rⁿ. Prove or disprove (if you think the following statement is false, give a counter-example and prove that your example works) that f(B) is bounded.

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