§3. Bolzano-Weierstrass and Cauchy's criterion
241
graph of cp reduces to a finite number of horizontal line segments and maybe
some isolated points. We also said that a function f defined on a bounded
interval I = (u, v) is regulated if it is the uniform limit of step functions on I
[see relation (11.2) of Chap. II], when we can give a meaning to the integral
of f on I immediately.
Corollary. Let f be a scalar function defined on a bounded interval I of JR.
Suppose that for any r > 0 there exists a step function cp on I such that
If(x) -cp(x)1 < r for all x E I {i.e. that f is regulated}. Then f has right and
left limit values at all points of I, and the set of its points of discontinuity is
countable.
It suffices to remark that a step function possesses left and right limits at
every point a E I and to apply the preceding theorem, replacing I by the set
X of x E I such that x < a (or x > a). Direct proof: for every step function cp
there exists an interval of the form la, a'[ with a' > a on which cp is constant;
the function f of the corollary is thus, for any r > 0, constant to within r
on an interval of the same type, whence the existence of f(a+) by Cauchy's
criterion.
Let (CPn) be a sequence of step functions converging uniformly to f and let
Dn be the (finite) set of discontinuities of CPn; the union D of Dn is countable
or finite (Chap. I). For a ~ D, all the CPn are continuous at a; similarly so
is f, qed.
We shall see Ii propos integration (Chap. V, nO 7) that, for I compact,
the existence of left and right limits chamcterises the regulated functions. In
particular, continuous and monotone functions are regulated. For this reason
we shall now extend the definition of a regulated function to the case of an
arbitrary interval I by requiring that it should possess left and right limits
at all points of I. A regulated function on I will then be a uniform limit of
step functions on all compact intervals K C I, but not necessarily on all of I.
Compact convergence again ...
13 - Passing to the limit in a series of functions
The results of the preceding n° can be translated into the language of series
of functions. Theorem 16 yields the following result:
Theorem 17 (Passage to the limit under the L sign). Let T be a
subset of C, let a be an adherent point of T and let L u(n, t) be a series of
junctions defined on T. Suppose that
(i) for all n the function t ~ u(n, t) tends to a limit u(n) as t tends to a,
(ii) the series L u(n, t) converges normally in T.
Then the sum s(t) of the given series tends to a limit when t tends to a, and
241
graph of cp reduces to a finite number of horizontal line segments and maybe
some isolated points. We also said that a function f defined on a bounded
interval I = (u, v) is regulated if it is the uniform limit of step functions on I
[see relation (11.2) of Chap. II], when we can give a meaning to the integral
of f on I immediately.
Corollary. Let f be a scalar function defined on a bounded interval I of JR.
Suppose that for any r > 0 there exists a step function cp on I such that
If(x) -cp(x)1 < r for all x E I {i.e. that f is regulated}. Then f has right and
left limit values at all points of I, and the set of its points of discontinuity is
countable.
It suffices to remark that a step function possesses left and right limits at
every point a E I and to apply the preceding theorem, replacing I by the set
X of x E I such that x < a (or x > a). Direct proof: for every step function cp
there exists an interval of the form la, a'[ with a' > a on which cp is constant;
the function f of the corollary is thus, for any r > 0, constant to within r
on an interval of the same type, whence the existence of f(a+) by Cauchy's
criterion.
Let (CPn) be a sequence of step functions converging uniformly to f and let
Dn be the (finite) set of discontinuities of CPn; the union D of Dn is countable
or finite (Chap. I). For a ~ D, all the CPn are continuous at a; similarly so
is f, qed.
We shall see Ii propos integration (Chap. V, nO 7) that, for I compact,
the existence of left and right limits chamcterises the regulated functions. In
particular, continuous and monotone functions are regulated. For this reason
we shall now extend the definition of a regulated function to the case of an
arbitrary interval I by requiring that it should possess left and right limits
at all points of I. A regulated function on I will then be a uniform limit of
step functions on all compact intervals K C I, but not necessarily on all of I.
Compact convergence again ...
13 - Passing to the limit in a series of functions
The results of the preceding n° can be translated into the language of series
of functions. Theorem 16 yields the following result:
Theorem 17 (Passage to the limit under the L sign). Let T be a
subset of C, let a be an adherent point of T and let L u(n, t) be a series of
junctions defined on T. Suppose that
(i) for all n the function t ~ u(n, t) tends to a limit u(n) as t tends to a,
(ii) the series L u(n, t) converges normally in T.
Then the sum s(t) of the given series tends to a limit when t tends to a, and