Math205a Fall Term 2002

Assignment 6

1. Let I ⊂ ℝ be an interval and assume that f : I → ℝ is monotone. Show that f is continuous if f(I) is also an interval.

2. Let f ∈ C(K, ℝ) for some compact set K ⊂ ℝ and assume that f > 0. Show that 1/f is uniformly continuous.

3. Let D_{f}, D_{g} ⊂ ℝ, f ∈ C(D_{f}, ℝ), g ∈ C(D_{g}, ℝ) and assume that g(D_{g}) ⊂ D_{f}. Show that f ∘ g ∈ C(D_{g}, ℝ).

4. Let x_{0} ∈ D ⊂^{o} ℝ and f ∈ C^{1}(D, ℝ). Prove that

a_{x_{0}} : D → ℝ , x ↦ f(x_{0}) + f′(x_{0})(x − x_{0})

is the best affine approximation to f as x → x_{0}.

The Homework is due Monday, November 18