Exercise 4.2:14 below will show that the above result can be looked at describing an ‘‘infinite dimensional space-filling curve’’.
4: R 18 A function that is continuous at all irrationals and discontinuous at all rationals (d : 3)
... Answers to True/False question 4.4:0. (a)T. (b)F.
Exercise not in Rudin: 4.5:0. Say whether each of the following statements is true or false.
(a) The function f (x) = 1 ⁄ x has a discontinuity of the second kind at x = 0.
(b) The function f such that f (x) = 1 ⁄ x for x ≠ 0 and f (0) = 0 has a discontinuity of the second kind at x = 0.
(c) Suppose f , g are real-valued functions on R such that f (x) = g(x) for all x ≠ 0, while f (0) ≠ g(0). Then if f is continuous at 0, g must have a discontinuity of the first kind at 0.
(d) Suppose g is a real-valued functions on R which is continuous for x ≠ 0 and has a discontinuity of the first kind at x = 0. Then there is a real-valued function f on R such that f (x) = g(x) for all x ≠ 0, and f is everywhere continuous.
4.6. MONOTONIC FUNCTIONS. (pp.95-97)
Relevant exercises in Rudin: 4:R15. Every continuous open map R→ R is monotonic. (d : 3)
If you don’t see how to get started on this one, you might look at f (x) = x2, an example of a map R → R which is not monotonic, and observe, by applying it to the segment ( –1, 1), that it is also not open. Try to show that every map that fails to be monotonic also fails to be open for the same sort of reason.
Exercises not in Rudin: 4.6:0. Say whether each of the following statements is true or false.
(a) Every constant function R→ R is both monotonically increasing and monotonically decreasing. (b) If f : R→ R is any function, then {x∈R! f is discontinuous at x} is a closed set.
4.6:1. Increasing functions have a property like uniform continuity. (d : 4)
Suppose α : [a, b] → R is a monotonically increasing function. Show that for every ε > 0 there exists a δ > 0 such that whenever a≤ x < y ≤ b with y – x < δ, either
α(y) –α(x) < ε, or there exists z∈[x, y] such that
α(z+) – α(z –) ≥ ε, and (α(y) – α(z+)) + (α(z –) – α(x)) < ε,
where if z = a we replace α(z–) by α(z) in the above inequalities, while if z = b we replace α(z+) by α(z).
4.7. INFINITE LIMITS AND LIMITS AT INFINITY. (pp.97-98) Relevant exercises in Rudin: None
Exercises not in Rudin: 4.7:0. Say whether each of the following statements is true or false.
(a) If f is a continuous real-valued function on a bounded segment (a, b), then as x→ b, either f (x) approaches a real number, or it approaches +
∞
, or it approaches –∞
.(b) Suppose f is a function R → R such that as x → +
∞
, f (x) either approaches a real number, or approaches +∞
, or approaches –∞
. Then as x→ +∞
, 1 ⁄ ( f (x)2+ 1) approaches a real number.(c) If f : R → R satisfies limx→+
∞
f (x) = a for some real number a, then the sequence of real numbers ( f (n)) converges to a.(d) If f : R → R is a continuous function, and if the sequence of real numbers ( f (n)) converges to the real number a, then limx→+
∞
f (x) = a.(e) Suppose f , g : R → R are functions such that limx→+
∞
f (x) = +∞
and limx→+∞
g(x) = –∞
. Then limx→+∞
f (x) + g(x) does not exist.(f ) Suppose f , g : R → R are functions such that limx→+
∞
f (x) = +∞
and limx→+∞
g(x) = –∞
. Then limx→+∞
f (x) + g(x) exists.4.7:1. A function on R that has limits at ±
∞
is uniformly continuous. (d : 2)Let f : R → R be a continuous function such that limx→+
∞
f (x) and limx→–∞
f (x) both exist. Show that f is uniformly continuous.(Note that, following Rudin’s conventions, the statement that the limits both exist means that they are real numbers. If f approached +
∞
, as x approached +∞
, for instance, Rudin would write ‘‘f (x) → +∞
as x → +∞
’’, but he would not write ‘‘limx→+∞
f (x) = +∞
’’, and would not say that limx→+∞
f (x) exists.)4.7:2. A continuous map asymptotic to a uniformly continuous map is uniformly continuous. (d : 2, 3) For what we want to prove, we first need to strengthen Theorem 4.19:
(a) Let f : X → Y be a continuous map between metric spaces, and K a compact subset of X. Prove that for every ε> 0 there exists a δ> 0 such that for all p∈K and x∈X with d ( p, x) < δ, one has d ( f ( p), f (x)) < ε. (If you wish, you may simply note, in precise detail, changes in the proof of Theorem 4.19 that will yield the above result.)
(b) If f, g : X → Y are two maps between metric spaces, let us say f and g are ‘‘asymptotic’’ to one another if for every ε > 0, the set {x∈X ! d( f (x), g(x)) ≥ ε} is contained in a compact subset of X. Show that any continuous map which is asymptotic to a uniformly continuous map is uniformly continuous. Then show that the result of 4.7:1 above follows from this result.
4.7:3. Cauchy criteria for limits of functions. (d : 2)
(a) Suppose f is a complex-valued function on a neighborhood of +
∞
, (c, +∞
). Show that limx→+∞
f (x) exists if and only if only if for every ε> 0 there exists a real number M≥c such that for all x, y∈(M, +∞
) one has | f (x) – f (y)| < ε.State a similar criterion for a complex-valued function defined on a neighborhood of –
∞
to have a limit as x→ –∞
. (Since the proof is almost identical, I don’t ask you to write it out.)(b) Suppose f is a complex-valued function defined on a subset E of a metric space X, and let p be a limit point of E in X. State and prove an analogous ‘‘Cauchy criterion’’ for limx→p f (x) to exist. (c) Show by example(s) that one or more of the above criteria fail if ‘‘complex-valued function’’ is replaced by ‘‘function into a metric space Y ’’. For which metric spaces Y will those results hold?
Chapter 5. Differentiation.
5.1. THE DERIVATIVE OF A REAL FUNCTION. (pp.103-106) Relevant exercise in Rudin: 5:R7. Something that looks like L’Hospital’s rule. (d : 1)
We won’t see L’Hospital’s rule itself until section 5.4. This exercise, despite its appearance, can be done using the definition of derivative, without calling on that result.
5:R13(a-d ). Behavior of |x|asin ( | x |– c). (d : 2)
In this exercise, for xasin ( | x |– c) read |x|asin ( | x |– c), since we have not defined what it means to raise a negative number to a non-integer power. In doing this, you may assume standard facts on how the function xr behaves as x → 0 through positive values; for which real numbers r it is unbounded, for which r it is bounded, and for which r it approaches 0, and likewise you may assume standard formulas for the derivatives of trigonometric functions and functions xr.
Parts (e-g) of the exercise should, strictly speaking, wait till section 5.5, when derivatives of higher order are defined, but since the definition is presumably familiar to everyone, the whole exercise might be
... Answers to True/False question 4.5:0. (a) F. (b) T. (c) T. (d) F. Answers to True/False question 4.6:0. (a) T. (b)F.
assigned here. It might also be assigned with section 5.3, in view of its relation to the topic of that section. One could at this point likewise look at the first question asked in 5:R21, about getting an f which is differentiable; but since the remaining parts of the question refer to higher-order differentiability, I have left that question to section 5.5.
Exercise not in Rudin: 5.1:0. Say whether each of the following statements is true or false.
(a) If a bounded function R→ R is continuous, then f′(t) exists for all real numbers t. (b) If f is a differentiable real-valued function on [0, 2] and limt→1 f (t) = 7, then f (1) = 7.
(c) If f and g are real-valued functions on [a, b], then at any point where f and g are both differentiable, f + g is also differentiable.
(d) If f and g are real-valued functions on [a, b], then at any point where f and f + g are both differentiable, g is also differentiable.
5.2. MEAN VALUE THEOREMS. (pp.107-108)
Relevant exercises in Rudin: 5:R1. A condition for f to be constant. (d : 2)