Real Analysis, Math 821.

Instructor: Dmitry Ryabogin

Assignment X.

1. Problem 1.

a) Is it possible to construct f(x), x ∈ [0, 1], such that f′(x) = D(x)? Here D(x) = 1 for x ∈ [0, 1] ∩ Q, and D(x) = 0 for x ∈ [0, 1] ∩ R \ Q. Hint: Prove the Darboux Theorem: If f(x) is differentiable on [0, 1], then ∀C ∈ [f′(0), f′(1)] there exists x ∈ [0, 1] such that f′(x) = C.

b) Construct a function f(x) on [0, 1] such that f′(x) exists at every x ∈ [0, 1], (and bounded), but f′(x) is not continuous for every x ∈ F, where F ⊂ [0, 1], and m(F) > 0. Hint: Let F ⊂ [0, 1], m(F) > 0 be closed, nowhere dense, and such that inf F = 0, sup F = 1. Define

f(x) = (x − a_{n})^{2}(x − b_{n})^{2} sin \frac{1}{(b_{n} − a_{n})(x − a_{n})(x − b_{n})}, x ∈ (a_{n}, b_{n}),

where [0, 1] \ F = ∪_{i=1}^{∞}(a_{i}, b_{i}), and f(x) = 0 otherwise.

c) Construct a continuous function f on R which is not differentiable at any point. Hint: Put ϕ(x) = x, x ∈ [0, 1], and ϕ(x) = 2 − x, x ∈ [1, 2]. Define ϕ_{0}(x) := ϕ(x), x ∈ [0, 2], and ϕ_{0}(x + 2) = ϕ_{0}(x). Then, define f(x) := ∑_{n=0}^{∞}(3/4)^{n}ϕ_{0}(4^{n}x).

a) Fix any x ∈ R and any m ∈ N. Observe that there exists k ∈ Z such that k ≤ 4^{m}x ≤ k + 1, and put α_{m} := 4^{−m}k, β_{m} := 4^{−m}(k + 1). Prove that |ϕ_{0}(4^{n}β_{m}) − ϕ_{0}(4^{n}α_{m})| = 0, for n > m, and |ϕ_{0}(4^{n}β_{m}) − ϕ_{0}(4^{n}α_{m})| = 4^{n−m} for n ≤ m.

b) Conclude that |f(β_{m}) − f(α_{m})| ≥ 1/2(3/4)^{m}, and show that f is not differentiable at x.

2. Problem 2. a) Let f(0) = 0, f(1) = 5, f(x) = 1 − x, for x ∈ (0, 1). Use definition to find the total variation of f(x) on [0, 1].

b) Write out f(x) = cos^{2} x on [0, π] as a difference of two increasing functions.

c) Let f(x) = x^{2}, x ∈ [0, 1), f(x) = x + 3, x ∈ (1, 2], f(1) = 5. Check that V_{0}^{2}(f) = V_{0}^{1}(f) + V_{1}^{2}(f). Write out f(x) as a difference of two increasing functions.

3. Problem 3. a) Let f : V_{0}^{1}(|f|) < ∞. Is it true that V_{0}^{1}(f) < ∞ ?

b) Let f be continuous on [0, 1], and such that V_{0}^{1}(|f|) < ∞. Prove that V_{0}^{1}(f) < ∞. Hint: Use the mean-value theorem.

4. Problem 4. a) Construct a continuous f(x) on [a, b] such that V_{a}^{b}(f) < ∞, but f(x) “is not Holder” for any α > 0, (f is said to satisfy the Holder condition for some α > 0 on [a, b], if there exists a constant K > 0 such that

∀x, y ∈ [a, b], |f(x) − f(y)| ≤ K|x − y|^{α}).

Hint: Take [a, b] = [0, 1/2], and f(x) = −1/ log x, x ∈ (0, 1/2], f(0) = 0.

b)* Construct an example of a continuous f(x) on [a, b], such that V_{a}^{b}(f) = ∞, but f “is Holder” of the given 0 < α < 1.

Hint: Let (a_{i})_{i=1}^{∞} be such that a_{i} > a_{i+1} > 0 and ∑_{i=1}^{∞} a_{i} = A. Put f(x) = 0 ∀x = a_{1}, a_{1}+

a_{2}, a_{1} + a_{2} + a_{3}, ...,; f(x) = 1/n at the point a_{1} + a_{2} + a_{+}... + a_{n−1} + a_{n}/2, n = 1, 2, 3, ...;

f(1) = 0, and make f to be linear on any segment of the type [ ∑_{i=1}^{n−1} a_{i}, ∑_{i=1}^{n−1} a_{i} + a_{n}/2],

[ ∑_{i=1}^{n−1} a_{i} + a_{n}/2, ∑_{i=1}^{n} a_{i}], and on the segments [0, a_{1}/2], [a_{1}/2, a_{1}].

To show that f “is Holder” of the given 0 < α < 1, take a_{n} := n^{−1/α}, and consider two cases, 1) points M_{1}(x_{1}, y_{1}), M_{2}(x_{2}, y_{2}) belong to “the same” part of the graph of f(x),

2) points M_{1}(x_{1}, y_{1}), M_{2}(x_{2}, y_{2}) do not belong to “the same” part of the graph of f(x).

5. Problem 5. a) Let (f_{n}(x))_{n=1}^{∞} be a sequence of functions having bounded variation

on [a, b]. Assume also that ∑_{n=1}^{∞} V_{a}^{b}(f_{n}) < ∞, and f_{n}(a) = 0, ∀n ∈ N. Prove that the

series ∑_{n=1}^{∞} f_{n}(x) is convergent ∀x ∈ [a, b], and V_{a}^{b}( ∑_{n=1}^{∞} f_{n}) ≤ ∑_{n=1}^{∞} V_{a}^{b}(f_{n}).

b) Let (f_{n}(x))_{n=1}^{∞} be a sequence of continuous functions having bounded variation on

[a, b]. Assume also that the series ∑_{n=1}^{∞} f_{n}(x) converges uniformly on [a, b]. Is it true

that V_{a}^{b}( ∑_{n=1}^{∞} f_{n}) < ∞?

Hint: Consider (f_{n}(x))_{n=1}^{∞} on [0, 1], f_{n}(x) := sin(nπ(x(n+1)−1))/n on [1/(n+1), 1/n], f_{n}(x) := 0 on [0, 1] \ [1/(n + 1), 1/n]. You may also use an example from Problem 1, c).

c) Construct a function f, which is of bounded variation on any finite segment (and hence is a difference of two monotonic functions), but, nevertheless, is not monotonic on any segment.

Hint: Let ϕ_{0}(x) = |x| for x ∈ [−1/2, 1/2], ϕ_{0}(x + 1) = ϕ_{0}(x), and let ϕ_{n}(x) :=

min(ϕ_{0}(x), 8^{−n}). Consider f(x) := ∑_{n=0}^{∞} 2^{−n}ϕ_{n}(8^{n}x).