§2. Uniform convergence
205
§2. Uniform convergence
5 - Limits of continuous functions
When a sequence or a series of functions fn(x) converges at all points x of
the set X on which they are defined one says that they converge simply; at
first glance this is the weakest possible concept of convergence for a sequence
of functions, but others, weaker and more subtle, have been invented, mainly
for the needs of the theory of integration.
As we saw at the end of nO 2 the square waves series raises a serious
problem, one which does not arise for finite sums: that of finding conditions
to ensure that a sequence fn(x) of functions defined and continuous on a
set X E JR, and converging simply, will have a limit function that is again
continuous on X. An answer is easy to find and rests on two principles better
stated naively:
(a) If two functions f and g are almost equal at a point a and if they
are almost constant on a neighbourhood of a, then they are almost
equal on a neighbourhood of a;
(b) If f and g are almost equal on a neighbourhood of a and if g is
almost constant on a neighbourhood of a, then f is almost constant
on a neighbourhood of a.
Let us also propose a more concrete preliminary exercise to the reader.
Two motorists in Giganes 11.2 W24 192-valve Spitzenracers drive south
along the autoroute du Soleil, several lengths apart, on the lane corresponding to their political convictions. Never having heard that the set of
speeds authorised (resp. tolerated) on a French motorway theoretically has
130 (resp. 180) kmh as a relatively strict upper bound, the leader drives
constantly at 220 kmh, to within 20 kmh. The second ensures that the
speeds of the two vehicles remain equal to within 10 kmh. What can one
say about the range of his speed?
Returning to the problem above, suppose that limfn(x) = f(x) exists for
all x E X and that f is continuous at a EX. Choose an r > 0 and consider
an n sufficiently large that
(5.1)
d [f(a), fn(a)] < r.
Since f is continuous at a,
(5.2)
d[f(a), f(x)] < r on a neighbourhood of a.
Since fn is continuous at a, similarly
(5.3)
d [fn(a), fn(x)] < r on a neighbourhood of a.
205
§2. Uniform convergence
5 - Limits of continuous functions
When a sequence or a series of functions fn(x) converges at all points x of
the set X on which they are defined one says that they converge simply; at
first glance this is the weakest possible concept of convergence for a sequence
of functions, but others, weaker and more subtle, have been invented, mainly
for the needs of the theory of integration.
As we saw at the end of nO 2 the square waves series raises a serious
problem, one which does not arise for finite sums: that of finding conditions
to ensure that a sequence fn(x) of functions defined and continuous on a
set X E JR, and converging simply, will have a limit function that is again
continuous on X. An answer is easy to find and rests on two principles better
stated naively:
(a) If two functions f and g are almost equal at a point a and if they
are almost constant on a neighbourhood of a, then they are almost
equal on a neighbourhood of a;
(b) If f and g are almost equal on a neighbourhood of a and if g is
almost constant on a neighbourhood of a, then f is almost constant
on a neighbourhood of a.
Let us also propose a more concrete preliminary exercise to the reader.
Two motorists in Giganes 11.2 W24 192-valve Spitzenracers drive south
along the autoroute du Soleil, several lengths apart, on the lane corresponding to their political convictions. Never having heard that the set of
speeds authorised (resp. tolerated) on a French motorway theoretically has
130 (resp. 180) kmh as a relatively strict upper bound, the leader drives
constantly at 220 kmh, to within 20 kmh. The second ensures that the
speeds of the two vehicles remain equal to within 10 kmh. What can one
say about the range of his speed?
Returning to the problem above, suppose that limfn(x) = f(x) exists for
all x E X and that f is continuous at a EX. Choose an r > 0 and consider
an n sufficiently large that
(5.1)
d [f(a), fn(a)] < r.
Since f is continuous at a,
(5.2)
d[f(a), f(x)] < r on a neighbourhood of a.
Since fn is continuous at a, similarly
(5.3)
d [fn(a), fn(x)] < r on a neighbourhood of a.