2008 Mathematics Bridge Program – Rochus Boerner – Tuesday, July 15th, 2008.
Mapping Properties of Continuous Real Functions
Recall the definition of continuity: 𝑓 is continuous at 𝑥 if and only if for any given (output accuracy)
𝜀 > 0, there is an (input accuracy) 𝛿 > 0 such that for any 𝑦 in the domain of 𝑓 with |𝑥 − 𝑦| < 𝛿, the
inequality |𝑓(𝑥) − 𝑓(𝑦)| < 𝜀 holds.
Write out complete proofs of the following theorems and turn them in.
Theorem 1: let (𝑎𝑛 ) ⊂ ℝ be a convergent sequence with 𝑎𝑛 → 𝑎. Suppose 𝑓 is a function defined at
least on the set 𝐴 = {𝑎𝑛 : 𝑛𝜖ℕ} ∪ {𝑎} and suppose 𝑓 is also continuous at 𝑎. Then 𝑓(𝑎𝑛 ) → 𝑓(𝑎).
Theorem 2: If 𝐴 ⊂ ℝ is open and 𝑓: 𝐴 → ℝ continuous, then for any open 𝒪 ⊂ ℝ, 𝑓 −1 (𝒪) is open as
well.
Give an example of how theorem 2 may fail if the assumption that 𝐴 ⊂ ℝ is open is dropped.
Theorem 3: If 𝑓: 𝐴 → ℝ is continuous, and 𝐾 ⊂ 𝐴 is compact, then 𝑓(𝐾) is also compact.
Write two different proofs of this theorem:
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One that assumes the sequential compactness definition.
One that assumes the Heine-Borel property as the definition.
Theorem 4: A continuous function maps intervals to intervals.
Prove this using the following definition of interval: a set 𝐼 ⊂ ℝ is an interval if it has the property that
for any 𝑥, 𝑦 𝜖 𝐼, 𝑥 < 𝑦, any number 𝑧 with 𝑥 < 𝑧 < 𝑦 is also contained in 𝐼. Put differently, a set 𝐼 ⊂ ℝ
is an interval if 𝑥, 𝑦 𝜖 𝐼, 𝑥 < 𝑦, implies [𝑥, 𝑦] ⊂ 𝐼.
Theorem 5: A continuous function maps closed intervals to closed intervals.
Is it true that a continuous function maps open intervals to open intervals? Prove or disprove.
Theorem 6: A continuous and strictly increasing function maps open intervals to open intervals.
Theorem 7: A continuous function on a closed interval assumes a maximum and a minimum.
Written solutions for all theorems are due Monday July 21st.