194
III - Convergence: Continuous variables
IRe(z) - Re(a) I = IRe(z - a)1 :::; Iz - al·
On putting z = x + iy, the functions z f--+ xPyq are similarly continuous
for all p, q EN, hence so is every function of z that can be expressed in
terms of polynomials with coefficients in C in the two real variables x and
y, hence also every function of the form h(z) = p(x, y)/q(x, y) where p and
q are polynomials. But here the points to exclude, defined by the equation
q(x, y) = 0, are not necessarily finite in number as in the case of lR: the function x 2 / (x 2 - y2 -1) is defined only off the equilateral hyperbola x 2 - y2 = 1.
This example is interesting in that it shows us that the quotient p/q is
defined on an open subset of C. This is a perfectly general fact: if f is a
function defined and continuous on lR (resp. q then the relation f(x) -=I- 0
defines an open subset of lR (resp. q. If indeed f(a) -=I- 0, we again have
f(x) -=I- 0 on a neighbourhood of a, as we saw in Theorem 2; the point a is
thus interior to the set f -=I- O.
More generally: for any open V in lR (resp. C), the inverse image
f-l(V) = U is open in lR (resp. C). For, given a E U and b = f(a) E V,
there exists an open ball B(b,r) C V; since f is continuous, there exists an
open ball B(a,r') which f maps into B(b,r); thus B(a,r') C U, qed. This
result remains valid for a function defined on an arbitrary set X C lR (for
example) on condition that one calls a subset U of X open in X if it possesses
the following property: for all a E U, U contains all the x E X (not lR or q
that are sufficiently close to a. All this generalises immediately (and becomes
clearer) in the framework of "metric spaces" , described in the Appendix.
The analytic functions form another class of continuous functions on Co
This is immediate. From the definition of analytic functions (Chap. II, n° 19),
f(a + h) is, for Ihl small, a power series in h, with constant term f(a); by
Chap. II, (14.7), we thus have
f(a + h) - f(a) = O(lhl),
Ihl --> 0,
from which continuity is clear.
Every reasonable algebraic operation (i.e. excluding division by 0) performed on continuous functions yields a continuous function again (Theorem 2). Another rather important operation, though not algebraic, possesses
the same property:
Theorem 3 (continuity of composite functions). Let X and Y be
two subsets of C, let a be a point of X, and b a point of Y; let f be a
map of X into Y such that f(a) = band 9 a map ofY into Co Suppose that
f is continuous at the point a and 9 is continuous at the point b. Then the
composite junction
h = 9 0 f : x t--+ g[f(x)]
is continuous at a.
III - Convergence: Continuous variables
IRe(z) - Re(a) I = IRe(z - a)1 :::; Iz - al·
On putting z = x + iy, the functions z f--+ xPyq are similarly continuous
for all p, q EN, hence so is every function of z that can be expressed in
terms of polynomials with coefficients in C in the two real variables x and
y, hence also every function of the form h(z) = p(x, y)/q(x, y) where p and
q are polynomials. But here the points to exclude, defined by the equation
q(x, y) = 0, are not necessarily finite in number as in the case of lR: the function x 2 / (x 2 - y2 -1) is defined only off the equilateral hyperbola x 2 - y2 = 1.
This example is interesting in that it shows us that the quotient p/q is
defined on an open subset of C. This is a perfectly general fact: if f is a
function defined and continuous on lR (resp. q then the relation f(x) -=I- 0
defines an open subset of lR (resp. q. If indeed f(a) -=I- 0, we again have
f(x) -=I- 0 on a neighbourhood of a, as we saw in Theorem 2; the point a is
thus interior to the set f -=I- O.
More generally: for any open V in lR (resp. C), the inverse image
f-l(V) = U is open in lR (resp. C). For, given a E U and b = f(a) E V,
there exists an open ball B(b,r) C V; since f is continuous, there exists an
open ball B(a,r') which f maps into B(b,r); thus B(a,r') C U, qed. This
result remains valid for a function defined on an arbitrary set X C lR (for
example) on condition that one calls a subset U of X open in X if it possesses
the following property: for all a E U, U contains all the x E X (not lR or q
that are sufficiently close to a. All this generalises immediately (and becomes
clearer) in the framework of "metric spaces" , described in the Appendix.
The analytic functions form another class of continuous functions on Co
This is immediate. From the definition of analytic functions (Chap. II, n° 19),
f(a + h) is, for Ihl small, a power series in h, with constant term f(a); by
Chap. II, (14.7), we thus have
f(a + h) - f(a) = O(lhl),
Ihl --> 0,
from which continuity is clear.
Every reasonable algebraic operation (i.e. excluding division by 0) performed on continuous functions yields a continuous function again (Theorem 2). Another rather important operation, though not algebraic, possesses
the same property:
Theorem 3 (continuity of composite functions). Let X and Y be
two subsets of C, let a be a point of X, and b a point of Y; let f be a
map of X into Y such that f(a) = band 9 a map ofY into Co Suppose that
f is continuous at the point a and 9 is continuous at the point b. Then the
composite junction
h = 9 0 f : x t--+ g[f(x)]
is continuous at a.