§ 1. Direct construction
327
Since we have always XO = 1, we deduce from the rules (1) that
(1.4)
and, since Xl = X,
(1.5)
rsimilarly, a result which, for s = n (an integer), justifies the expression xl/n
for nth roots. Formula (5) also shows that, for s rational and not just an
integer, the maps f : X f--+ X S and 9 : x f--+ Xl/s are mutually inverse.
Finally we note that 1 n = 1 for all nEZ, so also
(1.6)
1 s = 1 for all s E iQ.
Ifhe power functions x f--+ xS, defined for the moment only for s E iQ, are
~continuous on the set x > 0 where they are defined and, for s nonzero,
strictly monotone.
Continuity is established as follows: if s = alb one composes the map
x f--+ xa, clearly continuous since a is integer, with the map x f--+ x l / b , inverse
to x f--+ x b and so also continuous, by the general result of Chap. III, n° 4,
Theorem 7; and the composition of continuous maps is a continuous function.
The fact that the functions x f--+ xn are strictly monotone for n E Z
nonzero, and therefore their inverses x f--+ xl/n are too, shows similarly that
the function X S is strictly increasing for s > 0 and decreasing for s < O.
·a - Definition of real powers
;Instead of considering X S as a function of x for a given s E iQ one can also fix
.an a > 0 and consider the function s f--+ as on iQ. Except for a = lone again
obtains a strictly monotone function; to be precise:
{
as < at
if 1 < a
(2.1)
s < t ===} as > at
if 0 < a < 1.
On putting t = s + u, so that at = aSa u , it is enough to see that
(2.2)
u > 0 ===} aU> 1 if a > 1 or aU < 1 if a < 1.
But since 1 = 1 u, (2) follows from the fact that the function x f--+ XU is strictly
increasing for u > 0, as we saw above.
After these preliminaries, let us move on to the definition of X S for s real.
It would be regrettable if the relations (1) were not to extend to arbitrary
real exponents. We need to prove the following result:
327
Since we have always XO = 1, we deduce from the rules (1) that
(1.4)
and, since Xl = X,
(1.5)
rsimilarly, a result which, for s = n (an integer), justifies the expression xl/n
for nth roots. Formula (5) also shows that, for s rational and not just an
integer, the maps f : X f--+ X S and 9 : x f--+ Xl/s are mutually inverse.
Finally we note that 1 n = 1 for all nEZ, so also
(1.6)
1 s = 1 for all s E iQ.
Ifhe power functions x f--+ xS, defined for the moment only for s E iQ, are
~continuous on the set x > 0 where they are defined and, for s nonzero,
strictly monotone.
Continuity is established as follows: if s = alb one composes the map
x f--+ xa, clearly continuous since a is integer, with the map x f--+ x l / b , inverse
to x f--+ x b and so also continuous, by the general result of Chap. III, n° 4,
Theorem 7; and the composition of continuous maps is a continuous function.
The fact that the functions x f--+ xn are strictly monotone for n E Z
nonzero, and therefore their inverses x f--+ xl/n are too, shows similarly that
the function X S is strictly increasing for s > 0 and decreasing for s < O.
·a - Definition of real powers
;Instead of considering X S as a function of x for a given s E iQ one can also fix
.an a > 0 and consider the function s f--+ as on iQ. Except for a = lone again
obtains a strictly monotone function; to be precise:
{
as < at
if 1 < a
(2.1)
s < t ===} as > at
if 0 < a < 1.
On putting t = s + u, so that at = aSa u , it is enough to see that
(2.2)
u > 0 ===} aU> 1 if a > 1 or aU < 1 if a < 1.
But since 1 = 1 u, (2) follows from the fact that the function x f--+ XU is strictly
increasing for u > 0, as we saw above.
After these preliminaries, let us move on to the definition of X S for s real.
It would be regrettable if the relations (1) were not to extend to arbitrary
real exponents. We need to prove the following result: