328
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
Theorem 1. Let a be a strictly positive real number. Then there is one and
only one monotone map f : JR ---+ JR't such that
(2.3)
f(s) = as for all SEQ.
The function f is continuous and it is the only continuous function satisfying (3).
Let us assume that a > 1, the case where a < 1 reducing to it by the
formula as = (l/a)-S and the case a = 1 being trivial. The desired function
f must clearly be increasing as it is already so on Q.
To establish existence and uniqueness, we first remark that, if f exists,
we must have
(2.4)
aU::; f(s) ::; a V for u,v E Q, u < s < v.
If we consider the increasing function s ~ as on Q (and not on JR ... ), then
Chap. III, nO 3 shows that it has left and right limits at all s E JR, which we
denote by2
(2.5)
f-(s) = lim aU, I+(s) = lim aV,
u-s-O
v_s+o
uEQ
vEQ
where the expression u ~ s - 0 means that u tends to s while remaining < s.
These limits are also least upper bounds and greatest lower bounds. With
this notation, the relation (4) is equivalent to
(2.6)
f-(s) ::; f(s) ::; I+(s)
since f(s) must majorise all the aU for u < sand minorise all the a V for
v> s.
In conclusion, we do not yet know f(s) for s real nonrational, but we do
know bounds between which f(s) must lie, namely the expressions (5). If we
can prove them equal, then (6) will determine f(s) without any ambiguity:
for then
f(s) = f-(s) = I+(s).
Now if u and v are rational numbers such that u < s < v, then definition
(5) shows on the one hand that
(2.7)
2 It is clear that f _ (8) and f+ (8) are precisely the left and right limits of the desired
function f, in the sense of Chap. III, nO 3. But this assumes that the function f
has been constructed. To avoid logical ambiguities we adopt this notation to
denote numbers which, at the end of the proof, will indeed be the left and right
limits f(8+) and f(8-), in fact equal to f(8), of the function f that we are now
beginning to construct.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
Theorem 1. Let a be a strictly positive real number. Then there is one and
only one monotone map f : JR ---+ JR't such that
(2.3)
f(s) = as for all SEQ.
The function f is continuous and it is the only continuous function satisfying (3).
Let us assume that a > 1, the case where a < 1 reducing to it by the
formula as = (l/a)-S and the case a = 1 being trivial. The desired function
f must clearly be increasing as it is already so on Q.
To establish existence and uniqueness, we first remark that, if f exists,
we must have
(2.4)
aU::; f(s) ::; a V for u,v E Q, u < s < v.
If we consider the increasing function s ~ as on Q (and not on JR ... ), then
Chap. III, nO 3 shows that it has left and right limits at all s E JR, which we
denote by2
(2.5)
f-(s) = lim aU, I+(s) = lim aV,
u-s-O
v_s+o
uEQ
vEQ
where the expression u ~ s - 0 means that u tends to s while remaining < s.
These limits are also least upper bounds and greatest lower bounds. With
this notation, the relation (4) is equivalent to
(2.6)
f-(s) ::; f(s) ::; I+(s)
since f(s) must majorise all the aU for u < sand minorise all the a V for
v> s.
In conclusion, we do not yet know f(s) for s real nonrational, but we do
know bounds between which f(s) must lie, namely the expressions (5). If we
can prove them equal, then (6) will determine f(s) without any ambiguity:
for then
f(s) = f-(s) = I+(s).
Now if u and v are rational numbers such that u < s < v, then definition
(5) shows on the one hand that
(2.7)
2 It is clear that f _ (8) and f+ (8) are precisely the left and right limits of the desired
function f, in the sense of Chap. III, nO 3. But this assumes that the function f
has been constructed. To avoid logical ambiguities we adopt this notation to
denote numbers which, at the end of the proof, will indeed be the left and right
limits f(8+) and f(8-), in fact equal to f(8), of the function f that we are now
beginning to construct.