21.9. EXERCISES 569

In particular, |t− s|= 12 j+1 so 21/p |t− s|1/p = 2− j/p

| fε (t)− fε (s)− ( fε′ (t)− fε′ (s))|= 2(

21/p)|t− s|1/p

which shows that

sup0≤s<t≤1

| fε (t)− fε′ (t)− ( fε (s)− fε′ (s))||t− s|1/p ≥ 21/p (2)

Thus there exists a set of uncountably many functions in C1/p ([0,T ]) and for any two ofthem f ,g, you get

∥ f −g∥C1/p([0,1]) > 2

so C1/p ([0,1]) is not separable.

21.9 Exercises1. Is N a Gδ set? What about Q? What about a countable dense subset of a complete

metric space?

2. ↑ Let f : R→ C be a function. Define the oscillation of a function in B(x,r) byωr f (x) = sup{| f (z)− f (y)| : y,z ∈ B(x,r)}. Define the oscillation of the functionat the point, x by ω f (x) = limr→0 ωr f (x). Show f is continuous at x if and onlyif ω f (x) = 0. Then show the set of points where f is continuous is a Gδ set (tryUn = {x : ω f (x) < 1

n}). Does there exist a function continuous at only the rationalnumbers? Does there exist a function continuous at every irrational and discontinu-ous elsewhere? Hint: Suppose D is any countable set, D = {di}∞

i=1, and define thefunction, fn (x) to equal zero for every x /∈ {d1, · · · ,dn} and 2−n for x in this finiteset. Then consider g(x)≡ ∑

∞n=1 fn (x). Show that this series converges uniformly.

3. Let f ∈C([0,1]) and suppose f ′(x) exists. Show there exists a constant, K, such that| f (x)− f (y)| ≤ K|x− y| for all y ∈ [0,1]. Let Un = { f ∈C([0,1]) such that for eachx ∈ [0,1] there exists y ∈ [0,1] such that | f (x)− f (y)| > n|x− y|}. Show that Un isopen and dense in C([0,1]) where for f ∈C ([0,1]),

∥ f∥ ≡ sup{| f (x)| : x ∈ [0,1]} .

Show that ∩nUn is a dense Gδ set of nowhere differentiable continuous functions.Thus every continuous function is uniformly close to one which is nowhere differen-tiable.

4. ↑ Suppose f (x) = ∑∞k=1 uk (x) where the convergence is uniform and each uk is a

polynomial. Is it reasonable to conclude that f ′ (x) = ∑∞k=1 u′k (x)? The answer is no.

Use Problem 3 and the Weierstrass approximation theorem to show this.

5. Let X be a normed linear space. A ⊆ X is “weakly bounded” if for each x∗ ∈X ′, sup{|x∗(x)| : x ∈ A} < ∞, while A is bounded if sup{∥x∥ : x ∈ A} < ∞. ShowA is weakly bounded if and only if it is bounded.