Continuous functions with constant multiplicity.
a) Construct a continuous function f: ℝ → ℝ. such that every real number occurs as the image of exactly three numbers.
b) (+) Let f: ℝ → ℝ be continuous. Suppose that each z ∊ ℝ occurs as the image of exactly k numbers. Prove that k must be odd. (Hint: Try to draw the graph of such a function with k even. For k even, use the Intermediate Value Theorem and the Maximum-Minimum Theorem to complete a proof by contradiction.)
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