§1. Direct construction
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To do this one chooses sequences (sn) and (tn) of mtional numbers which
converge to sand t. Then
by rule (I) for rational exponents, qed.
The rule
(II)
can be established in two stages.
Suppose first that t E Q and choose a sequence of numbers Sn E Q which
converges to s. Then snt converges to st and in consequence we have
(3.1)
since the exponential functions are continuous. Moreover, and for the same
reason,
X S = limxSnj
now we already know that the "power function" y f-+ yt is continuous on JR+
for t E Qj it follows that
(XS)t
(lim xSn)t = lim [(X Sn )t] (continuity of y f-+ yt for t E Q) =
lim(xsnt) (because Sn, t E Q) = x st
by (1), whence (II) for s E JR and t E Q.
The general case remains. It is enough to approximate t by rational tn
and to pass to the limit in the relation
since, for s given, the function t f-+ (XS)t is continuous, being an exponential
function of t, the left hand side tends to (XS)tj since stn converges to st, the
right hand side tends, for the same reason, to xst. Whence (II) in the general
case.
Rule (II) shows in particular that, for any nonzero s E JR,
(III)
so that the maps x f-+ X S and x f-+ x 1 / 8 of JR+ onto JR+ are mutually inverse
bijections.
We have still to establish the formula
(IV)
Here again it is enough to approximate s by Sn E Q and to apply (IV) to
the Snj the continuity of the exponential functions immediately validates the
passage to the limit.