QuestionQuestion

2. a) Suppose f: E -> Y where E is a subset of a metric space and Y is a metric space, and that f is uniformly continuous. Show: If {xₙ} is a Cauchy sequence in E, then {f(xₙ)} is a Cauchy sequence in Y.
b) Suppose f: (a, b) -> R, f uniformly continuous, that {xₙ}, {x̄ₙ} are sequences in (a, b) with xₙ -> b, x̄ₙ -> b, f(xₙ) -> y, f(x̄ₙ) = y̅. Show y̅ = u.
c) Combine a), completeness of R, b) [with the analog when xₙ -> a, x̄ₙ -> a] to show:
If f is uniformly continuous on (a, b), then there exist ya, yb ∈ R so that if g: [a, b] -> R, g(x) = { f(x), a < x < b and ya, x = a, and yb, x = b } then g is continuous.

3. Define f: [0, 3] -> R by f(x) = {1, if 0 ≤ x ≤ 1 and 2, if 1 < x ≤ 2 and 3, if 2 < x ≤ 3}
Show that f is Riemann integrable (using only the definition of integrability and find ∫f(x) dx)

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