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on its own terms.
Bishop called his version of radical analysis ˜Constructive Analysis™ and
the reader will ¬nd an excellent account of it in the ¬rst few chapters of [7].
The ¬rst serious attempt to found a radical analysis was due to Brouwer who
called it ˜Intuitionism™.
Appendix I


This appendix consists of short notes on various topics which I feel should
be mentioned but which did not ¬t well into the narrative structure of this

Compactness When mathematicians generalised analysis from the study of
metric spaces to the study of more general ˜topological spaces™, they needed a
concept to replace the Bolzano-Weierstrass property discussed in this book.
After some experiment, they settled on a property called ˜compactness™.
It can be shown that a metric space has the property of compactness
(more brie¬‚y, a metric space is compact) when considered as a topologi-
cal space if and only if it has the Bolzano-Weierstrass property. (See Ex-
ercises K.196 and K.197 if you would like to know more.) We showed in
Theorem 4.2.2 that a subset of Rn with the standard metric has the Bolzano-
Weierstrass property if and only if it is closed and bounded.
Thus, if you read of a ˜compact subset E of Rn ™, you may translate
this as ˜a closed and bounded subset of Rn ™ and, if you read of a ˜com-
pact metric space™, you may translate this as ˜a metric space having the
Bolzano-Weierstrass property™. However, you must remember that a closed
bounded metric space need not have the Bolzano-Weierstrass property (see
Exercise 11.2.4) and that, for topological spaces, the Bolzano-Weierstrass
property is not equivalent to compactness.

Abuse of language The language of mathematics is a product of history as
well as logic and sometimes the forces of history are stronger than those of
logic. Logically, we need to talk about the value sin x that the function sin
takes at the point x, but, traditionally, mathematicians have talked about the
function ˜sin x™. To avoid this problem mathematicians tend to use phrases
like ˜the function f : R ’ R given by f (x) = x2 ™ or ˜the map x ’ x2 ™. In


Section 13.3, when we wish to talk about the map x ’ L(», x) = t(x)’»f (x)
we write ˜L(», ) = t ’ »f ™. (Many mathematicians dislike leaving a blank
space in a formula and use a place holder ˜·™ instead. They write ˜L(», ·) =
t(·) ’ »f (·)™.)
To a 19th century mathematician and to most of my readers this may
appear unnecessary but the advantage of the extra care appears when we need
to talk about the map x ’ 1. However, from time to time, mathematicians
revert to their traditional habits.
Bourbaki calls such reversions ˜abuses of language without which any
mathematical text runs the risk of pedantry, not to say unreadability.™
Perhaps the most blatant abuse of language in this book concerns se-
quences. Just as f (x) is not a function f , but the value of f at x, so an is
not the sequence

a1 , a 2 , a 3 , a 4 . . . ,

but the nth term in such a sequence. Wherever I refer to ˜the sequence
an ™, I should have used some phrase like ˜the sequence (an )∞ ™. Perhaps
future generations will write like this, but it seemed to me that the present
generation would simply ¬nd it distracting.

Non-uniform notation When Klein gave his lectures on Elementary Mathe-
matics from An Advanced Standpoint [28] he complained

There are a great many symbols used for each of the vector op-
erations and, so far, it has proved impossible to produce a gener-
ally accepted notation. A commission was set up for this purpose
at a scienti¬c meeting at Kassel (1903). However, its members
were not even able to come to a complete agreement among them-
selves. None the less, since their intentions were good, each mem-
ber was willing to meet the others part way and the result was to
bring three new notations into existence1 ! My experience in such
things inclines me to the belief that real agreement is only possi-
ble if there are powerful economic interests behind such a move.
. . . But there are no such interests involved in vector calculus and
so we must agree, for better or worse, to let every mathematician
cling to the notation which he ¬nds most convenient or “ if he is
dogmatically inclined “ the only correct one.
In later editions Klein recorded the failure of similar committees set up at the Rome
International Mathematical Congress (1908).
Please send corrections however trivial to twk@dpmms.cam.ac.uk

Most mathematics students believe that there should be a unique notation
for each mathematical concept. This is not entirely reasonable. Consider the
derivative of a function f : R ’ R. If I am interested in the slope of the
curve y = f (x) it is natural to use the suggestive Leibniz notation . If I
am interested in the function which is the derivative of f it is is more natural
to consider f , and if I am interested in the operation of di¬erentiation rather
than the functions it operates on I may well write Df . Di¬erent branches of
mathematics may have genuine reasons for preferring di¬erent notations for
the same thing.
Even if she disagrees, the student must accept that di¬erent notations
exist and there is nothing that she can do about it. A quick visit to the
library reveals the following notations for the same thing (here f : R3 ’ R
is a well behaved function) :-

, fxxz , fxxz , Dxxz f, f,113 , f113 , f113 , D113 f, D(2,0,1) f, D(2,0,1) f
2 ‚z
and several others.
Here are three obvious pieces of advice.
(1) If you are writing a piece of mathematics and several notations exist
you must make it clear to the reader which one you are using.
(2) If you are reading a piece of mathematics which uses an unfamiliar
notation try to go along with it rather than translating it back into your
favourite notation. The advantage of a familiar notation is that it carries you
along without too much thought, the advantage of an unfamiliar notation is
that it makes you think again ” and, who knows, the new notation might
turn out to better than the old.
(3) Do not invent a new notation when a reasonably satisfactory one
already exists.

Left and right derivatives In many ways, the natural subsets of Rm to do
analysis on are the open sets U , since, given a function f : U ’ Rn and a
point x ∈ U we can then examine the behaviour of f ˜in all directions round
x™, on a ball

B(x, ) = {y : y ’ x < },

for some, su¬ciently small, > 0.
However, as we saw in Theorem 4.3.1 and Theorem 4.5.5, continuous
functions behave particularly well on closed bounded sets. This creates a
certain tension, as the next exercise illustrates.

Exercise I.1. (i) Show that the map Rm ’ R given by x ’ x is contin-
(ii) If E is a closed bounded set, show that there exists an e0 ∈ E such
that e0 ≥ x for all x ∈ E. Give an example to show that e0 need not be
(iii) Show that the only subset of Rm which is both open and closed and
bounded is the empty set.
This tension is often resolved by considering functions de¬ned on an open
set U , but working on closed bounded subsets of U . In the special case, when
we work in one dimension and deal with intervals, we can use a di¬erent trick.
De¬nition I.2. Let b > a and consider f : [a, b] ’ R. We say that f has
right derivative f (a+) at a if
f (a + h) ’ f (a)
’ f (a+)
as h ’ 0 through values h > 0.
Exercise I.3. De¬ne the left derivative f (b’), if it exists.
When mathematicians write that f is di¬erentiable on [a, b] with deriva-
tive f they are using the following convention.
De¬nition I.4. If f : [a, b] ’ R is di¬erentiable at each point of (a, b) and
has right derivative f (a+) at a and left derivative f (b’) at b, we say that
f is di¬erentiable on [a, b] and write f (a) = f (a+), f (b) = f (b’).
Observe that this is not radically di¬erent from our other suggested ap-
Lemma I.5. If f : [a, b] ’ R is di¬erentiable on [a, b] we can ¬nd an every-
˜ ˜
where di¬erentiable function f : R ’ R with f (t) = f (t) for t ∈ [a, b].
Proof. Set
f (t) = f (a) + f (a)(t ’ a) for t < a,
˜ for a ¤ t ¤ b,
f (t) = f (t)
f (t) = f (b) + f (b)(t ’ b) for b < t.
It is easy to check that f has the required properties.
Exercise I.6. If f : [a, b] ’ R is di¬erentiable on [a, b] with continuous
derivative, show that we can ¬nd an everywhere di¬erentiable function f :
R ’ R with continuous derivative such that f (t) = f (t) for t ∈ [a, b].
Please send corrections however trivial to twk@dpmms.cam.ac.uk

If f : [a, b] ’ R is di¬erentiable on [a, b] and f : [a, b] ’ R is di¬eren-
tiable on [a, b], it is natural to say that f is twice di¬erentiable on [a, b] with
derivative f = (f ) , and so on.
Exercise I.7. If f : [a, b] ’ R is twice di¬erentiable on [a, b] with continuous
second derivative, show that we can ¬nd an everywhere twice di¬erentiable
˜ ˜
function f : R ’ R with continuous second derivative such that f (t) = f (t)
for t ∈ [a, b]. Generalise this result.

Piecewise de¬nitions Occasionally mathematicians de¬ne functions as piece-
wise continuous, piecewise continuously di¬erentiable and so on. The under-
lying idea is that the graph of the function is made up of a ¬nite number of
well behaved pieces.
De¬nition I.8. A function f : [a, b] ’ R is piecewise continuous if we can
a = x0 < x1 < · · · < x n = b
and continuous functions gj : [xj’1 , xj ] ’ R such that f (x) = gj (x) for all
x ∈ (xj’1 , xj ) and all 1 ¤ j ¤ n
De¬nition I.9. A function f : [a, b] ’ R is piecewise linear (respectively
di¬erentiable, continuously di¬erentiable, in¬nitely di¬erentiable etc.) if it
is continuous and we can ¬nd
a = x0 < x1 < · · · < x n = b
such that f |[xj’1 ,xj ] : [xj’1 , xj ] ’ R is linear (respectively di¬erentiable, con-
tinuously di¬erentiable, in¬nitely di¬erentiable etc.) for all 1 ¤ j ¤ n.
Notice that De¬nition I.8 does not follow the pattern of De¬nition I.9.
Exercise I.10. (i) Show by means of an example that a piecewise continuous
function need not be continuous.
(ii) Show that a piecewise continuous function is bounded.
Exercise I.11. (This is a commentary on Theorem 8.3.1.) (i) Show that
any piecewise continuous function f : [a, b] ’ R is Riemann integrable.
(ii) Show that, if f : [a, b] ’ R is continuous on (a, b) and bounded on
[a, b], then f is Riemann integrable.
The de¬nitions just considered are clearly rather ad hoc. Numerical ana-
lysts use a more subtle approach via the notion of a spline (see, for example
Chapters 18 and onwards in [42]).
Appendix J

Executive summary

The summary is mainly intended for experts but may be useful for revision.
It may also be more useful than the index if you want to track down a
particular idea. Material indicated ™ . . . ™ or ™™ . . . ™™ is not central
to the main argument. Material indicated [ . . . ] is in appendices or exercises;
this material is either not central to the main argument or is such that most
students will have met it in other courses.

Introduction to the real number system
Need for rigorous treatment of analysis (p. 1). Limits in R, subsequences,
sums and products (p. 3). Continuity of functions from R to R (p. 7). The
real numbers R form an ordered ¬eld obeying the fundamental axiom that
every increasing bounded sequence converges (p. 9). Axiom of Archimedes
(p. 10). [Decimal expansion (Exercise 1.5.12, p. 13).] The intermediate value
theorem, proof by lion hunting (p. 14). [Countability (Appendix B, p. 383).
The real numbers are uncountable (Exercise 1.6.7, p. 17). Cantor™s proof of
the existence of transcendentals (Exercise B.7, p. 385). Explicit construction
of a transcendental number (Exercise K.12, p. 435).] Di¬erentiation and the
mean value inequality (one dimensional case) (p. 18). Intermediate value
theorem equivalent to fundamental axiom (p. 22). ™™Further informal
discussion of the status of the fundamental axiom (p. 25).™™

Equivalents of the fundamental axiom
Supremum, existence for bounded non-empty sets equivalent to fundamental
axiom, use as proof technique (p. 31). Theorem of Bolzano-Weierstrass,
equivalent to fundamental axiom, use as proof technique (p. 37).

Higher dimensions
Rm as an inner product space, Cauchy-Schwarz and the Euclidean norm


(p. 43). Limits in Rm (p. 46). Theorem of Bolzano-Weierstrass in Rm (p. 47).
Open and closed sets (p. 48). Theorem of Bolzano-Weierstrass in the context
of closed bounded subsets of Rm (p. 49). Continuity for many dimensional
spaces (p. 53). The image of a continuous function on a closed bounded
subset of Rm is closed and bounded, a real-valued continuous function on a
closed bounded subset of Rm is bounded and attains its bounds (p. 57). [The
intersection of nested, non-empty, closed, bounded sets is non-empty (Exer-
cise 4.3.8, p. 59).] Rolle™s theorem and the one dimensional mean value theo-
rem (p. 60). Uniform continuity, a continuous function on a closed bounded
subset of Rm is uniformly continuous (p. 64).

(This material is treated in Rm .) General principle of convergence (p. 66).
Absolute convergence implies convergence for sums (p. 69). Comparison test
(p. 70). Complex power series and the radius of convergence (p. 71). ™Ratio
test and Cauchy™s condensation test (p. 70). Conditional convergence, al-
ternating series test, Abel™s test, rearrangement of conditionally convergent
series (p. 78). Informal discussion of the problem of interchanging limits
(p. 81). Dominated convergence theorem for sums, rearrangement of abso-
lutely convergent series, Fubini™s theorem for sums (p. 84). The exponential
function mainly for R but with mention of C, multiplication of power series.
(p. 91). The trigonometric functions, notion of angle (p. 98). The logarithm
on (0, ∞), problems in trying to de¬ne a complex logarithm (p. 102). Powers
(p. 109). Fundamental theorem of algebra (p. 113).™

Di¬erentiation from Rn to Rm
Advantages of geometric approach, de¬nition of derivative as a linear map,
Jacobian matrix (p. 121). Operator norm, chain rule and other elementary
properties of the derivative (p. 127). Mean value inequality (p. 136). Simple
local and global Taylor theorems in one dimensions, Cauchy™s example of
a function with no non-trivial Taylor expansion, Taylor theorems depend
on the fundamental axiom (p. 141). Continuous partial derivatives imply
di¬erentiability, symmetry of continuous second order derivatives, informal
treatment of higher order local Taylor theorems, informal treatment of higher
order derivatives as symmetric multilinear maps (p. 146). Discussion, partly
informal, of critical points, hill and dale theorem (p. 154).

Riemann integration
Need for precise de¬nition of integral and area, Vitali™s example (p. 169).
De¬nition of the Riemann integral via upper and lower sums, elementary
properties, integrability of monotonic functions (p. 172). Integrability of
Please send corrections however trivial to twk@dpmms.cam.ac.uk

continuous functions, fundamental theorem of the calculus, Taylor™s theorem
with integral form of remainder (p. 182). ™ Di¬erentiation under the inte-
gral for ¬nite range, Euler-Lagrange equation in calculus of variations, use
and limitations, Weierstrass type example (p. 190).™ Brief discussion of the
Riemann integral of Rm -valued functions, f¤ f (p. 202).
™ Class of Riemann integrable functions not closed under pointwise
convergence (p. 205). Informal discussion of improper Riemann integration
(p. 207). Informal and elementary discussion of multiple integrals (no change
of variable formula), Fubini for continuous functions on a rectangle (p. 212),
Riemann-Stieltjes integration (p. 217). Recti¬able curves and line integrals,
Schwarz™s example showing the problems that arise for surfaces (p. 224).™

Metric spaces
™ Usefulness of generalising notion of distance illustrated by Shannon™s
theorem on the existence of good codes (p. 233).™ Metric spaces, norms,
limits, continuity, open sets (p. 241). All norms on a ¬nite-dimensional
space are Lipschitz equivalent (p. 246). Continuity of functions between
normed spaces (p. 251). ™ Informal discussion of geodesics illustrated by
the Poincar´ metric on the upper half plane (p. 254).™

Complete metric spaces
De¬nition of completeness, examples of complete and incomplete metric
spaces including among incomplete ones the L1 norm on C([a, b]) (p. 263).
Completeness and total boundedness, equivalence of the conjunction of these
properties with the Bolzano-Weierstrass property (p. 272).
Uniform metric is complete, restatement of result in classical terms (uni-
form limit of continuous functions is continuous, general principle of uni-
form convergence) (p. 275). Uniform convergence, integration and di¬erenti-
ation, restatement for in¬nite sums, di¬erentiation under an in¬nite integral
(p. 282). Local uniform convergence of power series, power series di¬eren-
tiable term by term, rigorous justi¬cation of power series solution of di¬eren-
tial equations (p. 288). ™ An absolutely convergent Fourier series converges
to the appropriate function (p. 298).™

Contraction mapping theorem
Banach™s contraction mapping theorem (p. 303). Existence of solutions of
di¬erential equations by Picard™s method (p. 305). ™ Informal discussion
of existence and non-existence of global solutions of di¬erential equations
(p. 310). Green™s function solutions for second order linear di¬erential equa-
tions (p. 318).™
The inverse function theorem (p. 329). ™ The implicit function theorem

(p. 339). Lagrange multipliers and Lagrangian necessary condition (p. 347).
Lagrangian su¬cient condition, problems in applying Lagrange multiplier
methods (p. 353).™

Completion of metric spaces
Density, completion of metric space, inheritance of appropriate structures
such as inner product (p. 355). Proof of existence of completion (p. 362).
™ Informal discussion of construction of Z from N, Q from Z, C from R
(p. 364). Construction of R from Q (p. 369).™ ™™ Rapid, optimistic and
informal discussion of foundational issues (p. 375).™™
Appendix K


At an elementary level, textbooks consist of a little explanation (usually sup-
plemented by a teacher) and a large number of exercises of a routine nature.
At a higher level the number of of exercises decreases and the exercises be-
come harder and more variable in di¬culty. At the highest level there may be
no exercises at all. We may say that such books consist of a single exercise:-
read and understand the contents. Because I would be happy if students
treated my text in this manner I have chosen to put most of the exercises in
an appendix.
I suspect that readers will gain most by tackling those problems which
interest them. To help them make a choice, I have labeled them in the
following manner [2.1, P]. The number 2.1 tells you that Section 2.1 may
be relevant, and the letters have the meanings given below. Like many similar
labeling systems it is not entirely satisfactory.
‘ Follows on from the preceding question.
‘‘ Follows on from an earlier question.
S Rather shorter or easier than the general run of questions.
M Methods type question. Forget theoretical niceties and concentrate on
getting an answer.
M! Just try and get an answer by fair means or foul.
T This question leads you through a standard piece of theory.
T! This question leads you through a standard piece of theory but in a
non-standard way.
P Problem type question.
G Uses general background rather than material in this book.
H The result of this exercise is not standard.
H! The result of this exercise is highly non-standard. Only do this exercise
if you are really interested.


Figure K.1: Apostol™s construction.

Exercise K.1. (irrationality of 2.) [1.1, G, T] The reader presumably
knows the classic proof that the equation n2 = 2m2 has no non-zero integer
solutions (in other words, x2 = 2 has no solution in Q). Here are two others.
(i) Show that, if n2 = 2m2 , then

(2m ’ n)2 = 2(n ’ m)2 .

Deduce that, if n and m are strictly positive integers with n2 = 2m2 , we
can ¬nd strictly positive integers n and m with n 2 = 2m 2 and n < n.
Conclude that the equation n2 = 2m2 has no non-zero integer solutions.
(ii) Our second argument requires more thought but is also more powerful.
We use it to show that, if N is a positive integer which is not a perfect square,
then the equation x2 = N has no rational solution.
To this end we suppose that x is a positive rational with x2 = N . Explain
why we can ¬nd a least positive integer m such that mx is an integer and
why we can ¬nd an integer k with k + 1 > x > k. Set m = mx ’ mk and
show that m is an integer, that m x is an integer and that m > m ≥ 1. The
required result follows by contradiction. (This argument and its extensions
are discussed in [3].)
(iii) Apostol gave the following beautiful geometric version of the argu-
ment of part (i) (see Figure K.1). It will appeal to all fans of Euclidean
geometry and can be ignored by everybody else.

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