Math205b Winter Term 2003

Assignment 9

Let a < b ∈ ℝ and denote by P the generic partition

a = x_{0} < x_{1} < ⋯ < x_{n} = b

of the interval [a, b]. Let y_{k} denote any point in [x_{k−1}, x_{k}]. Finally define

△(P) = sup_{k=1,...,n}(x_{k} − x_{k−1})

the maximum interval length of the partition P.

1. Let

S([a, b], ℝ) = {φ : [a, b] → ℝ |

φ|_{[x_{k−1},x_{k}]} ≡ φ_{k} ∈ ℝ , k = 1, …, n for some P }

be the vector space of step functions defined on the interval [a, b]. Show that any f ∈ C([a, b], ℝ) can be approximated by a sequence of step functions, that is, ∀ε > 0 there is φ ∈ S([a, b], ℝ) s.t

∥f − φ∥_{∞} = sup_{x∈[a,b]} |f(x) − φ(x)| ≤ ε .

2. Let g ∈ C^{1}([a, b], ℝ) be increasing. Prove that

∫_{a}^{b} f(x) dg(x) = lim_{△(P)→0} ∑_{k=1}^{n} f(y_{k})(g(x_{k}) − g(x_{k−1}))

is well-defined for f ∈ C([a, b], ℝ). Also show that

∫_{a}^{b} f(x) dg(x) = ∫_{a}^{b} f(x)g′(x) dx .

3. Let f ∈ C^{1}([a, b], ℝ) and x, y ∈ [a, b]. Prove the mean value theorem in integral form

f(y) = f(x) + (y − x) ∫_{0}^{1} f′((1 − t)x + ty)dt .

4. Let x_{0} ∈ U_{x_{0}} ⊂^{o} ℝ and f, g ∈ C^{3}(U_{x_{0}}) with

f(x_{0}) = f′(x_{0}) = 0 = g′(x_{0}) = g(x_{0}) and g′′(x_{0}) ≠ 0 .

Show that f/g is differentiable at x_{0} and compute the derivative there.

5. You ask a question.

The Homework is due Friday January 17 2002