312
Appendix to Chapter III
More precisely: Let X, Y and Z be metric spaces, f a map of X into y
and 9 a map of Y into Z. Suppose that f is continuous at a E X and 9 is
continuous at f(a) = bEY. Then the map h = go f is continuous at a.
For let c = h(a) = g(b) and choose an r > O. Since 9 is continuous at b
there exists an r' > 0 such that
d(b,y) < r' ==} d[c,g(y)] < r.
Since f is continuous at a there is an r" > 0 such that
d(a,x) < r" implies d[b,f(x)] < r'.
It is clear that then the relation d(a,x) < r" implies d[c, h(x)] < r, qed.
One verifies immediately that if f' and f" are functions defined on metric
spaces X' and X", with values in metric spaces Y' and Y", and continuous
at a' E X' and a" E X", then the map
(x', x") f-----+ (f'(x'), f"(x"))
of X' x X" into Y' x Y" is continuous at (a', a"). If one has a further continuous map p : Y' x Y" ----+ Y into another metric space Y, then the map
(x', x") f-----+ p[f'(x'),f"(x")] is continuous at (a', a"), a vast and trivial generalisation of theorems on the sums, products, upper and lower envelopes,
etc. of continuous functions.
Similarly one can extend the concept of uniform convergence to the maps
of a metric space X into another metric space Y and show that a uniform
limit of continuous maps is again continuous. To do this, one defines the
distance between two maps f, 9 of X into Y by the formula
(4.1)
dx(f,g) = sup d[!(x), g(x)].
Apart from a detail of no importance in this context - the distance can be infinite -, all the properties of a distance are clearly satisfied. Then uniform convergence can be expressed, as in Chap. III, by the relation limdx (fn'!) = 0
or by the fact that, for any r > 0, there exists an integer N such that
n > N ==} d [fn(x), f(x)] $ r for any x E X.
The fact that a uniform limit of continuous functions is again continuous is
proved exactly as we proved it for X = Y = C.
Note moreover that one can replace X by any subset E of X in definition (1), and define the distance dE(f, g) on E, so also uniform convergence
onE.
If these abstract trivialities have forced the reader to reflect, of which I
am not convinced 52 , they will at least have served for that.
52 Maybe there will be readers who will ask why we did not develop the concepts
expounded in Chap. II and III in this general framework from the outset. Reply:
though they may appear trivial to you after you have read these chapters, they
would have seemed incomprehensible if you had met them at the outset.
Appendix to Chapter III
More precisely: Let X, Y and Z be metric spaces, f a map of X into y
and 9 a map of Y into Z. Suppose that f is continuous at a E X and 9 is
continuous at f(a) = bEY. Then the map h = go f is continuous at a.
For let c = h(a) = g(b) and choose an r > O. Since 9 is continuous at b
there exists an r' > 0 such that
d(b,y) < r' ==} d[c,g(y)] < r.
Since f is continuous at a there is an r" > 0 such that
d(a,x) < r" implies d[b,f(x)] < r'.
It is clear that then the relation d(a,x) < r" implies d[c, h(x)] < r, qed.
One verifies immediately that if f' and f" are functions defined on metric
spaces X' and X", with values in metric spaces Y' and Y", and continuous
at a' E X' and a" E X", then the map
(x', x") f-----+ (f'(x'), f"(x"))
of X' x X" into Y' x Y" is continuous at (a', a"). If one has a further continuous map p : Y' x Y" ----+ Y into another metric space Y, then the map
(x', x") f-----+ p[f'(x'),f"(x")] is continuous at (a', a"), a vast and trivial generalisation of theorems on the sums, products, upper and lower envelopes,
etc. of continuous functions.
Similarly one can extend the concept of uniform convergence to the maps
of a metric space X into another metric space Y and show that a uniform
limit of continuous maps is again continuous. To do this, one defines the
distance between two maps f, 9 of X into Y by the formula
(4.1)
dx(f,g) = sup d[!(x), g(x)].
Apart from a detail of no importance in this context - the distance can be infinite -, all the properties of a distance are clearly satisfied. Then uniform convergence can be expressed, as in Chap. III, by the relation limdx (fn'!) = 0
or by the fact that, for any r > 0, there exists an integer N such that
n > N ==} d [fn(x), f(x)] $ r for any x E X.
The fact that a uniform limit of continuous functions is again continuous is
proved exactly as we proved it for X = Y = C.
Note moreover that one can replace X by any subset E of X in definition (1), and define the distance dE(f, g) on E, so also uniform convergence
onE.
If these abstract trivialities have forced the reader to reflect, of which I
am not convinced 52 , they will at least have served for that.
52 Maybe there will be readers who will ask why we did not develop the concepts
expounded in Chap. II and III in this general framework from the outset. Reply:
though they may appear trivial to you after you have read these chapters, they
would have seemed incomprehensible if you had met them at the outset.