206
III - Convergence: Continuous variables
Starting from r /3 and using the quadrilateral inequality we obtain - principle (i) above - the following statement:
(5.4)
For every r > 0 and every sufficiently large n we have
d [f(x), fn(x)] < r on a neighbourhood of a.
This means, more exactly, that for every large n there exists an r~ > 0 such
that, for x E X,
d(x, a) < r~ ==* d [f(x), fn(x)] < r.
Suppose conversely that this condition is satisfied for any r > 0 and let
us show that then f is continuous at a. We choose an n large enough for f(x)
and fn(x) to be equal to within r on a neighbourhood of a and in particular
at the point a. Since f n is continuous, f n (x) and f n (a) are equal to within r
on a neighbourhood of a. The quadrilateral inequality - principle (ii) above
- then shows that d[f(a), f(x)] < 3r on a neighbourhood of a in X, qed.
When the condition (4) is satisfied one says that the convergence of the
sequence (In) to f is locally uniform at the point a, (dangerous terminology
because of the risk of confusion with the stronger concept of a sequence which
converges uniformly on a fixed neighbourhood of a .. . ). This is the necessary
and sufficient condition for continuity at a of the limit function f, but it is not
very handy, and in practice one uses two simpler properties, more restricted
than (4).
First and foremost is that of (global) uniform convergence on X. This
means that for every r > 0 there exists an integer N, independent of x, such
that
(5.5)
{(n > N) & (x E X)} ==* d [f(x), fn(x)] < r.
Condition (4) is then satisfied for all a E X and for all n > N without it
being necessary to impose the least restriction on d( x, a): r~ has no further
role, in other words you can, for n > N, choose it ad libitum. We have met
an example in Chap. II, at the end of nO 14: if a power series L anz n has
radius of convergence R > 0, then for all p < R there exist positive constants
M and q < 1 such that, for all n,
If(z) - sn(z)1 < Mqn
on the disc X : Izl :S p, where Sn denotes the nth partial sum of the series.
Since qn tends to 0, one thus has
If(z) - sn(z)1 < r for all z E X
provided that n is sufficiently large, Le. once n exceeds an integer N independent of z E x.
III - Convergence: Continuous variables
Starting from r /3 and using the quadrilateral inequality we obtain - principle (i) above - the following statement:
(5.4)
For every r > 0 and every sufficiently large n we have
d [f(x), fn(x)] < r on a neighbourhood of a.
This means, more exactly, that for every large n there exists an r~ > 0 such
that, for x E X,
d(x, a) < r~ ==* d [f(x), fn(x)] < r.
Suppose conversely that this condition is satisfied for any r > 0 and let
us show that then f is continuous at a. We choose an n large enough for f(x)
and fn(x) to be equal to within r on a neighbourhood of a and in particular
at the point a. Since f n is continuous, f n (x) and f n (a) are equal to within r
on a neighbourhood of a. The quadrilateral inequality - principle (ii) above
- then shows that d[f(a), f(x)] < 3r on a neighbourhood of a in X, qed.
When the condition (4) is satisfied one says that the convergence of the
sequence (In) to f is locally uniform at the point a, (dangerous terminology
because of the risk of confusion with the stronger concept of a sequence which
converges uniformly on a fixed neighbourhood of a .. . ). This is the necessary
and sufficient condition for continuity at a of the limit function f, but it is not
very handy, and in practice one uses two simpler properties, more restricted
than (4).
First and foremost is that of (global) uniform convergence on X. This
means that for every r > 0 there exists an integer N, independent of x, such
that
(5.5)
{(n > N) & (x E X)} ==* d [f(x), fn(x)] < r.
Condition (4) is then satisfied for all a E X and for all n > N without it
being necessary to impose the least restriction on d( x, a): r~ has no further
role, in other words you can, for n > N, choose it ad libitum. We have met
an example in Chap. II, at the end of nO 14: if a power series L anz n has
radius of convergence R > 0, then for all p < R there exist positive constants
M and q < 1 such that, for all n,
If(z) - sn(z)1 < Mqn
on the disc X : Izl :S p, where Sn denotes the nth partial sum of the series.
Since qn tends to 0, one thus has
If(z) - sn(z)1 < r for all z E X
provided that n is sufficiently large, Le. once n exceeds an integer N independent of z E x.