7 Continuous Functions
Let f be an JR-valued function defined on an open interval (a, b) of JR. In
passing to *JR, we may regard f as being defined for all hyperreal x between
a and b, since *(a, b)o= {x E *JRo: a< x < b}.
Cauchy's Account of Continuity
Informally, we describe the assertion
f is continuous at a point c in the interval (a, b)
as meaning that f(x) stays "close to" f(c) whenever xis "close to" c. The
way Cauchy put it in 1821 was that
the function f(x) is continuous with respect to x between the
given limits if between these limits an infinitely small increase
in the variable always produces an infinitely small increase in
the function.
From the enlarged perspective of *JR, this account can be made precise:
Theorem 7.1.1 f is continuous at the real point c if and only if f(x)
f(c) for all x E *JR such that x c, i.e., iff
f(hal(c)) hal(f(c)).
Proof The standard definition is that f is continuous at c iff for each
open interval (f (c) c, f (c) + c) around f (c) in lR there is a corresponding
-
76 7. Continuous Functions
open interval (c-8, c+ 8) around c that is mapped into (!(c)o-c, f(c) +c)
by f. Since a < c < b, the number 8 can be chosen small enough so that
the interval ( c -8, c + 8) is contained with (a, b), ensuring that f is indeed
defined at all points that are within 8 of c.
Continuity at c is thus formally expressed by the sentence
(Vc E R+) (:38 E JR+) (Vx E R) ( lxo-ci < 8n--+ if(x)o-/(c) I c.
• limx-+c-f(x) = L iff f(x) L for all x E *A with x c and x c f ( x) = f (c).
(3) Use infinitesimals to discuss the continuity of the following functions:
-
n
{ s
in if X :;f 0,
fl(x)
0nif X= 0.
sin if x # 0,
h(x) { xo
0 if X= 0.
if x is rational,
h(x) { 1
0nif x is irrational. { x
if x is rational,
-x if x is irrational.
{ 01 if X is irrational, fs(x) if x = 7; E Q in simplest form with n 2 1.
7.5 The Intermediate Value Theorem
This fundamental result of standard real analysis states that
if the real function f is continuous on the closed interval [a, b]
in JR, then for every real number d strictly between f (a) and
f(b) there exists a real c E (a, b) such that f(c) =d.
There is an intuitively appealing proof of this using infinitesimals. The basic
idea is to partition the interval [a, b] into subintervals of equal infinitesimal
width, and locate a subinterval whose end points have !-values on either
side of d. Then c will be the common shadow of these end points. In this
way we "pin down" the point at which the !-values pass through d.
We deal with the case f(a) < f(b), so that f(a) < d < f(b). First, for
each (limited) n E N, partition [a, b] into n equal subintervals of width
(bo-a)fn. Thus these intervals have end points Pk = a+ k(bo-a)fn for
0 ::; k ::; n. Then let sn be the greatest partition point whose !-value is less
than d. Indeed, the set
}
{Pk : f(Pk) < dxis finite and nonempty (it contains Po•= a but not Pn =ob). Hence Sn exists
as the maximum of this set, and is given by some Pk with k < n.
No for all n EN we have
aSSn < b and f(sn) < d::; f(sn + (bo-a)jn),
and so by transfer, these conditions hold for all n E *N.
To obtain an infinitesimal-width partition, choose an unlimited hypernat
ural N. Then SN is limited, as a::; BN < b, so has a shadow c = sh(sN) E JR
80 7. Continuous Functions
(by transfer, sN is a number of the form a+K(bo-a) IN for some K E *N).
But (b-a) IN is infinitesimal, so sN and sN + (b-a) IN are both infinitely
close to c. Since f is continuous at c and cis real, it follows (Theorem 7.1.1)
that f(sN) and f(sN + (bo-a)IN) are both infinitely close to f(c). But
f(sN) < d::; f(sN + (b-a)IN),
so d is also infinitely close to f(c). Since f(c) and d are both real, they
must then be equal. D
7.6 The Extreme Value Theorem
If the real function f is continuous on the closed interval [a, b]
in :IR, then f attains an absolute maximum and an absolute
minimum on [a, b], i.e., there exist real c, d E [a, b] such that
f(c)::; f(x) ::; f(d) for all x E [a, b].
Proof. To obtain the asserted maximum we construct an infinitesimalwidth
partition of [a, b], and show that there is a particular partition point
whose f-value is as big as any of the others. Then d will be the shadow
of this particular partition point. As with the intermediate value theorem,
the construction is first approximated by finite partitions with subintervals
of limited width l. In these cases there is always a partition point with
n
maximum f-value. Then transfer is applied.
For each limited n E N, partition [a, b] into n equal subintervals, with
end points a+ k(bo-a)ln for 0::; k ::; n. Then let sn E [a, b] be a partition
point at which f takes its largest value. In other words, for all integers k
such that 0 ::; k ::; n,
a::; Sn ::; b and f(a + k(b-a)/n) ::; f(sn)· (ii)
By transfer, (ii) holds for all n E *N and all hyperintegers k such that
0 ::; k ::; n.
Similarly to the intermediate value theorem, choose an unlimited hypernatural
N and put d = sh(sN) E JR. Then by continuity
(iii)
Now the "infinitesimal-width partition"
P ={ao+ k(b-a)INo: k E *Nnand 0::; k::; N}
has the important property that it provides infinitely close approximations
to all real numbers between a and b: the halo of each x E [a, b] contains
points from this partition. To show this, observe that if x is an arbitrary
real number in [a, b], then for each n EN there exists an integer k < n with
a+ k(b-a) In::; x ::; a+ (k + l)(b-a)ln.
7.7 Uniform Continuity
Hence by transfer there exists a hyperinteger K < N such that x lies in
the interval
[a+ K(bo-a)/N, a+ (K + 1)(bo-a)/N]
of infinitesimal width (bo-a)/N. Therefore x a+ K(bo-a)/N, so x is
indeed infinitely close to a member of P. It follows by continuity ofnf at x
that
f(x) f(a + K(bo-a)/N). (iv)
But the values of f on P are dominated by f(sN), as (ii) holds for all
n E *N, so
f(a + K(bo-a)/N) ::; f(sN). (v)
Putting (iii), (iv), and (v) together gives
f(x) f(a + K(bo-a)/N) ::; f(sN) f(d),
which implies f(x) ::; f(d), since f(x) and f(d) are real (Exercise 5.5(2)).
Thus f attains its maximum value at d.
The proof that f attains a minimum is similar. D
7.7 Uniform Continuity
If A 1R and f : A -----t .IR, then f is uniformly continuous on A if the
following sentence is true:
('Vs E JR+) (38 E JR.+) (Vx, yEA) (lxo-Yl < 8 -----t lf(x)o-f(y)l < s)
(compare this to the formal sentence just prior to Theorem 7.1.3). Essentially,
this says that for a givenns, the same 8 for the continuity condition
works at all points of A.
Theorem 7.7.1 f is uniformly continuous on A if and only if x y
implies f(x)o""' f(y) for all hyperreals x, y E *A.
Proof. Exercise. D
Theorem 7.7.1 displays the distinction between uniform and ordinary continuity
in a more intuitive and readily comprehensible way than the standard
definitions do. For by Theorem 7.1.1, f is continuous on A IR iff x y
implies f(x) "'of(y) for all x, y E *A withny standard. Thus uniform continuity
amounts to preservation of the "infinite closeness" relation at all
hyperreal points in the enlargement *A of A, while continuity only requires
preservation of this relation at the real points.
Of course for some sets, these two requirements come to the same thing:
Theorem 7. 7.2 If the real function f is continuous on the closed interval
[a, b] in IR, then f is uniformly continuous on [a, b].
82 7. Continuous Functions
Proof Take hyperreals x, y E *[a, b] with x y. Let c = sh(x). Then
since a x bnand x c, we have c E [a, b], and so f is continuous at
c. Applying Theorem 7.1.1, we get f(x) f(c) and f(y) f(c), whence
f(x) f(y). Hence f is uniformly continuous by Theorem 7.7.1. 0
7.8 Exercises on Uniform Continuity
(1) Explain why the argument just given fails for intervals (a, b), (a, b],
(a,+oo), (-oo,b), etc. that are not closed.
(2) Show that f(x) = 1/x is not uniformly continuous on (0, 1).
(3) If f is uniformly continuous on JR. and (sn : n E N) is a Cauchy
sequence, show that (f(sn) : n EN) is a Cauchy sequence.
(4) Let the real function f be monotonic on [a, b], and suppose that for
all real r between f(a) and f(b) there exists a real c E [a, b] such that
f(c) = r. Prove that f is continuous on [a, b].
The property of closed intervals that makes Theorem 7.7.2 work will be
examined further in Section 10.4 when we study compactness from the
hyperreal perspective.
7.9 Contraction Mappings and Fixed Points
A function f : JR. JR. is said to satisfy a Lipschitz condition if there is a
positive real constant c such that
lf(x)o-f(y)J c JxoYl (vi)
-
for all x, y E JR. Such a function is always continuous, indeed uniformly
continuous, as is readily explained by infinitesimal reasoning. First observe
that (vi) holds for all hyperreal x, y by transfer. But then if x y, we
have that cJxn-yJ is infinitesimal, since cis real and Jx-yJ 0, so by (vi)
if(x) -f(y)J is infinitesimal, making f(x) f(y). Hence f is uniformly
continuous by the characterisation of Theorem 7. 7.1.
A contraction mapping is a Lipschitz function with constant c less than
1. Such a function acts on any two points to move them closer to each
other. It turns out that a contraction mapping has a fixed point: a point x
satisfying f(x) = x. It certainly cannot have two such points, for if f(x) = x
and f(y) = y, then
Jx-Yl = if(x)o-f(y)J c Jx-yJ,
7.9 Contraction Mappings and Fixed Points
and since c < 1, this is possible only if lx-Yl = 0 and hence x = y.
Consider for example the contraction mapping f defined by
f(x) = 2
X + 2
1
(what is the constant c here?). Its fixed point is the unique solution to
+ = x, namely x = 1. Moreover, this fixed point can be approached by
starting at any real number x and repeatedly applying f to generate the
sequence
X 3 X 7
f(f(x)) = 4 + 4' f(f(f(x))) =
8 + B'
The nth term of this sequence is
X 1-+1--2n 2n('
so the sequence does indeed converge to 1 regardless of what x is (this can
also be effectively demonstrated visually by plotting a graph of the function
and the terms of the sequence).
Theorem 7.9.1 Any contraction mapping f: -IR has a (unique} fixed
point.
Proof Let c be the Lipschitz constant for f. Take any x E 'nput soo= x,
and inductively define
Observe that
ls1 -s2l
ls2o-s3l
ls3o-s41
Sn+l = f(sn)·
< clsoo-s1l,
< cols1o-s2l :S c2olsoo-s1!,
< c ls2o-s3l :S c3olsoo-s1l,
(vii)
and so on. In general, for n E N we get
Hence
<
(viii)
!soo-s1l +clsoo-s1l +c2olsoo-s1l + ··· +cn-lolsoo-s1i
lsoo-s11-en
1o-c lsoo-
l
(l + c + c2 + + cn-l)
· · ·
sll,
84 7. Continuous Functions
and therefore
(ix)
for all n EN.
The standard proof of this theorem uses the more general formulan·
to prove that the sequence (sn : n E N) is Cauchy, hence convergent, and
that its limit is a fixed point for f. Here we will instead extend to the
hypersequence (sn :E *N) and take the shadow of any term in the extended
tail.
Thus if n E *N is unlimited, then by transfer lso -snnl is bounded by
the real right side of (ix), so sn is limited and has a shadow L E JR. Then
Sn L, and so as f is continuous, f(sn) f(L). But f(sn) = Sn+l by
transfer of (vii), and sn+1 Sn by transfer of (viii), since en is infinitesimal
when c < 1 and n is unlimited, making lsn-sn+InI infinitesimal. Altogether
then we have
f(L) f(sn) = Sn+l Sn L,
giving f(L) L. Since f(L) and L are real, it follows that they are equal,
sonL is the desired fixed point. D
Notice that the fact that the sequence x, f(x), f(f(x)), ... converges to a
fixed point of f now becomes a consequence of our proof, rather than being
part of the proof as in the standard argument. For we have shown that for
any unlimited n, the shadow sh(sn) exists and is a fixed point. Since there
can be only one fixed point, it follows that all extended terms have the same
shadow, and hence (Theorem 6.1.1) that the original sequence converges
to this shadow.
Theorem 7.9.1 is an instance of the Banach fixed point theorem, which
asserts the existence of a fixed point for a contraction mapping on any
"complete metric space"n. Essentially the same nonstandard analysis can
be used for the proof in that more general setting.
7.10 A First Look at Permanence
One of the distinctive features of nonstandard analysis is the presence of
so-called permanence principles, which assert that certain functions must
exist, or be defined, on a larger domain than that which is originally used
to define them. For instance, any real function f : A --t lR automatically
extends to the enlargement *A of its real domain A.
In discussing continuity of a real function f at a real point c, we may
want (the extension of) f to be defined at points infinitely close to c. For
7.11 Exercises on Permanence of Functions
this it suffices that f be defined on some real neighbourhood (c-c-, c+c:) in
JR, for then the domain of the extension of I includes the enlarged interval
*( c-c:, c +c:), which contains the halo hal(c) of c. But the converse of this is
also true: if the extension of I is defined on hal(c), then I must be defined
on some real interval of the form (c -c:, c +c:), and hence on *( c-c:•, c+ c:).
In fact, for this last conclusion it can be shown that it suffices that I
be defined on some hyperreal interval ( c -d, c + d) of infinitesimal radius
d. This is our first example of a permanence statement that is sometimes
called Cauchy's principle. It asserts that if a property holds for all points
within some infinitesimal distance of c, then it must actually hold for all
points within some real (hence appreciable) distance of c. At present we
can show this for the transforms of properties expressible in the formal
language £fR.. If cp(x) is a formula of this language for which there is some
positive d 0 such that
*cp(x) is true for all hyperreal x with c-d < x < c + d,
then the sentence
(3y E *JR+) (Vx E *JR) ( lx-cl < Y *cp)
is seen to be true by interpreting y as d. But then by existential transfer
there is some real c > 0 such that
(Vx E JR) ( lxo-cl < c: y;),
so that cp is true throughout ( c-c, c +c) in JR. Hence by universal transfer
back to *JR,
(Vx E *JR) ( lx-cl < c *cp),
showing that
*cp(x) is true for all hyperreal x with c-c < x < c +c.
Note that in this argument cis a real number. Later it will be shown that
permanence works for any hyperreal number in place of c, and applies to
a much broader class of properties than those expressible in the language
LfR. (cf. Theorems 11.9.1 and 15.1.1).
7.11 Exercises on Permanence of Functions
(1) If f is a real function and c E ' verify in detail that f (x) is defined
for all x c if and only if I ( x) is defined for all real x in some open
interval (cn-c, c +oc) with real radius c > 0.
(2) Let I be a real function that is defined on some open neighbourhood
of c E JR. Show that if I is constant on hal(c), then it is constant on
some interval (c-c, c + c:) JR.
86 7. Continuous Functions
(3) Let f be a real function that is continuous on some interval Ao-If
f(x) is real for all x E *A, show with the help of the previous exercise
that f is constant on A.
7.12 Sequences of Functions
Let (f n : n E N) be a sequence of functions f n : A defined on some
subset A of R The sequence is said to converge pointwise to the function
f : A if for each x E A the -valued sequence Un(x) : n E N)
converges to the number f(x). Symbolically, this asserts that
('ilx E A) lim fn(x) = f(x),
n->oo
which is rendered in full by the sentence
('ilx E A) ('i/c E +) (:Jm EN) ('iln EN) (n > m !fn(x)o-f(x)! <
c).
In this statement, the integer m that is asserted to exist depends on the
choice of x E A as well as on c. More strongly, we say that (f n : n E N)
converges uniformly to the function f if m depends only on c in the sense
that for a given c, the same m works for all x E A:
('ilc E JR+) (:Jm EN) ('ilx E A) ('iln EN) (n > m !fn(x)o-f(x)! M we have xn < c and hence xn 0.
Now, this M will be unlimited, because when n is limited, x 1 implies
xn 1. Hence { xn : n E N} is contained entirely within the halo of 1.
But there is a permanence principle that concludes from this that there is
some unlimited N such that { xn : n N} is contained in the halo of 1
(cf. Robinson's sequential lemma in Section 15.2). In particular, xN '/:-0,
i.e., fN(x) '/:-f(x), showing that the condition of Theorem 7.12.2 is violated,
and therefore that the original standard sequence (fn : n E N) is not
uniformly convergent to f.
7.13 Continuity of a Uniform Limit
A sequence Un : n E N) of continuous functions can converge pointwise
to a discontinuous function. We have just discussed the standard example:
n
take fn(x) = xoon A= [0, 1]. Under uniform convergence this phenomenon
cannot occur. Here is a hyperreal approach to this classical result:
Theorem 7.13.1 If the functions Un : n EN) are all continuous on A
R, and the sequence converges uniformly to the function f : A -+ IR, then
f is continuous on A.
88 7. Continuous Functions
Proof Let c belong to A. To prove that f is continuous at c, we invoke
Theorem 7.1.3(2). If x E *A with x c, we want f(x) f(c), i.e., if(x)of(
c)i < c for any positive real c. The key to this is to analyse the inequality
if(x)o-f(c)i :S if(x)o-fn(x)i + lfn(x)o-fn(c)i + lfn(c)o-f(c)i. (x)
On the right side, the middle term lfn(x)o-fn(c)i will be infinitesimal for
any n E N because x "-J c and fn is continuous at c. By taking a large
enough n, the first and last terms on the right can be made small enough
that the sum of the three terms is less than c.
To see how this works in detail, for a given c E JR+ we apply the definition
of uniform convergence to the number c /4 to get that there is some integer
m E N such that
n > m implies lfn(x)o-f(x)i < c/4
for all n E N and all x E A, and hence for all n E *N and all x E *A by
universal transfer.
Now fix n as a standard integer, say by putting n = m + 1. Then for any
x E *A with x
coit follows, since x, c E *A, that
lfn(x)o-f(x)i, lfn(c)o-f(c)i < c/4,
and so in (x) we get
!f(x)o-f(c)i < c/4 + infinitesimal + c/4 < c
as desired.
Note that this proof is a mixture of standard and nonstandard arguments:
it uses the hyperreal characterisation of continuity of fn and f, but the
standard definition of uniform convergence of (f n : n E N) rather than the
characterisation given by Theorem 7.12.2.
In fact, we could invoke Theorem 7.12.2 to make the first and last terms
on the right of the inequality (x) become infinitesimal (instead of less than
c/4), by choosing n to be unlimited. But then what happens to the middle
term lfn(x)o-fn(c)i when n is unlimited? Can we constrain it to still be
infinitesimal? We will take this up first as a separate question.
7.14 Continuity in the Extended Hypersequence
Given a sequence Un : n E N) of functions that are continuous on a set
A IR, it is natural to wonder about the continuity properties of the
extended terms fn of the associated hypersequence (fn : n E *N).
Now, when n is unlimited, then fn is a function from *A to *IR, and
even when restricted to A it may take values that are not real, e.g., when
7.14 Continuity in the Extended Hypersequence 89
n
fn(x) = x. Thus fn may not be the extension of any -valued function,
and so we cannot apply the transfer argument of Theorem 7.1.1 directly.
However, we can demonstrate a continuity property of f n that corresponds
to that of 7.1.3(3). Thus even though fn may not map the whole of the
halo of a point y into the halo of f n (y), it will do this to some sufficiently
small infinitesimal neighbourhood of y:
Theorem 7.14.1 If the functions Un : n EN) are all continuous on A
' then for any n E *N and any y E *A there is a positive infinitesimal d
such that fn(x) fn(Y) for all x E *A with lx-Yl 0 such that
lx-cl < d implies fn(x) fn(c)
90 7. Continuous Functions
whenever x E *A. But now if x E *A and lx-cl