Reference no: EM13966685

1. If (K, d) is a compact metric space and u ∈ K , show that for any finite M and 0 α ≤ 1, { f ∈ Lip(α, M ): | f (u)|≤ M } is compact for dsup.

2. If S = [0, 1] with its usual metric and α > 1, show that Lip(α, 1) contains only constant functions. Hint: For 0 ≤ x ≤ x + h ≤ 1, f (x + h) - f (x ) = },1≤ j ≤n f (x + jh/n) - f (x +( j - 1)h/n). Give an upper bound for the absolute value of the j th term of the right, sum over j , and let n → ∞.

3. Find continuous functions fn from [0, 1] into itself where fn → 0 pointwise but not uniformly as n → ∞. Hint: Let fn (1/n) = 1, fn (0) ≡ fn (2/n) ≡ 0. (This shows why monotone convergence, fn ↓ f0, is useful in Dini's theorem.)

4. Show that the functions fn (x ) := xn on [0, 1] are not equicontinuous at 1, without applying any theorem from this section.