§l. The intermediate value theorem
193
(and, in fact, for functions more special than just continuous). Dirichlet's
function, equal to 1 for X rational and to 0 for x irrational, does not lend
itself to numerical analysis even though it was very interesting to the founders
of the set theory, who invented far more bizarre functions!; one finds good
approximations of them today in the business of the computer science, under
the name of "fractals" .
One can translate this fundamental definition into yet another language.
Let us say that a function defined onto a set M (and maybe elsewhere) is
constant to within r on M if
d[f(x'), f(x")] ::; r for any x', x" E M.
Continuity at a can then be expressed by saying that for all n EN, there
exists an open ball B with centre a such that f is constant to within lO-n on
X n B. This condition is clearly sufficient since it shows in particular that,
for all n, one has If(x) - f(a)1 ::; lO-n for all x E X sufficiently close to a. If,
conversely, f is continuous at a, then If(x) - f(a)1 ::; r/2 for all x E X n B,
where B is a suitably chosen ball with centre a, whence If(x') - f(x")1 ::; r
for all x', x" E X n B. We have seen, for example, in Chap. II (nO 4, 10,
14), that the functions xn (n E N) and expx are continuous on C and that
the function log x is continuous (and even differentiable) on R+. Theorem 1
provides the following result immediately:
Theorem 2. Let f and 9 be scalar functions defined on a set X C C and
let a be a point of X where f and 9 are continuous. Then the functions
f + 9 : X f-----> f(x) + g(x) and fg: X f-----> f(x)g(x)
are continuous at a. If g(a) i:- 0 then g(x) i:- 0 on a neighbourhood of a and
the function f /g : x 1---+ f(x)/g(x), defined for g(x) i:- 0, is continuous at a.
First consequence: on JR, every rational function hex) = p(x)/q(x), where p
and q are polynomials, is continuous at any x [of course, one excludes the
points where q(x) = 0]. Indeed, the constant functions and the function x 1---+ x
are trivially continuous; hence all monomials ax n , then every polynomial,
then every quotient of polynomials.
There is a similar statement for C. It is of course clear that the functions
Re(z) and Im(z) are continuous on C since, for example,
1 Let f be a function defined on an interval I, with complex values. If f is continuous, one can consider the point f ( t) of C as a point moving in the plane as a
function of time, which suggests the naive idea that the "trajectory" is a highly
regular curve "of one dimension". In 1890, a clever Italian, Giuseppe Peano, also
one of the creators of mathematical logic, demonstrated a quite simple construction of a trajectory which passes through all the points of a square, in other
words, a continuous and surjective map I ----> I x I; "bijective" is impossible.
The effect on the mathematicians was no less sensational than that of Weierstrass' non-differentiable continuous function mentioned later. See Hairer and
Wanner, Analysis by Its History, pp. 289 and 296 (pictures).
193
(and, in fact, for functions more special than just continuous). Dirichlet's
function, equal to 1 for X rational and to 0 for x irrational, does not lend
itself to numerical analysis even though it was very interesting to the founders
of the set theory, who invented far more bizarre functions!; one finds good
approximations of them today in the business of the computer science, under
the name of "fractals" .
One can translate this fundamental definition into yet another language.
Let us say that a function defined onto a set M (and maybe elsewhere) is
constant to within r on M if
d[f(x'), f(x")] ::; r for any x', x" E M.
Continuity at a can then be expressed by saying that for all n EN, there
exists an open ball B with centre a such that f is constant to within lO-n on
X n B. This condition is clearly sufficient since it shows in particular that,
for all n, one has If(x) - f(a)1 ::; lO-n for all x E X sufficiently close to a. If,
conversely, f is continuous at a, then If(x) - f(a)1 ::; r/2 for all x E X n B,
where B is a suitably chosen ball with centre a, whence If(x') - f(x")1 ::; r
for all x', x" E X n B. We have seen, for example, in Chap. II (nO 4, 10,
14), that the functions xn (n E N) and expx are continuous on C and that
the function log x is continuous (and even differentiable) on R+. Theorem 1
provides the following result immediately:
Theorem 2. Let f and 9 be scalar functions defined on a set X C C and
let a be a point of X where f and 9 are continuous. Then the functions
f + 9 : X f-----> f(x) + g(x) and fg: X f-----> f(x)g(x)
are continuous at a. If g(a) i:- 0 then g(x) i:- 0 on a neighbourhood of a and
the function f /g : x 1---+ f(x)/g(x), defined for g(x) i:- 0, is continuous at a.
First consequence: on JR, every rational function hex) = p(x)/q(x), where p
and q are polynomials, is continuous at any x [of course, one excludes the
points where q(x) = 0]. Indeed, the constant functions and the function x 1---+ x
are trivially continuous; hence all monomials ax n , then every polynomial,
then every quotient of polynomials.
There is a similar statement for C. It is of course clear that the functions
Re(z) and Im(z) are continuous on C since, for example,
1 Let f be a function defined on an interval I, with complex values. If f is continuous, one can consider the point f ( t) of C as a point moving in the plane as a
function of time, which suggests the naive idea that the "trajectory" is a highly
regular curve "of one dimension". In 1890, a clever Italian, Giuseppe Peano, also
one of the creators of mathematical logic, demonstrated a quite simple construction of a trajectory which passes through all the points of a square, in other
words, a continuous and surjective map I ----> I x I; "bijective" is impossible.
The effect on the mathematicians was no less sensational than that of Weierstrass' non-differentiable continuous function mentioned later. See Hairer and
Wanner, Analysis by Its History, pp. 289 and 296 (pictures).