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IV - Powers, Exponentials, Logarithms, Trigonometric Functions
The reader will easily prove that in fact the relation
(6.5)
f(nx) = f(x)n for all x E lR. and all n E Z
is enough to characterise the exponential functions among the real continuous
or monotone functions.
Theorem 3. Every real-valued function defined on lR. which satisfies
f(x + y) = f(x) + f(y), and is monotone or continuous, is a linear function.
Put f(l) = a, and use (3) to show that f(x) = ax first for x E I'll, then
for x E Z, then for x E Q. The function being monotone or continuous, the
formula extends immediately to arbitrary real values of x.
Here, as above, the relation f(nx) = n.f(x) leads to the same result 6 . Note
further that this remains valid if f has complex values; but one must then
assume f continuous since the other possible hypothesis is then meaningless.
Theorem 4. Every real-valued function, defined for x > 0, which is not
identically zero, satisfies f(xy) = f(x)f(y), and is monotone or continuous,
is a power function.
Such a function in fact has positive values since f(x) = f(X 1 / 2 X 1 / 2 ) =
f(x 1 / 2 )2. It can never vanish, since f(c) = 0 would imply that f(x)
f(c)f(x/c) = 0 for all x > o.
Now let us choose an a> 0, not 1, and put g(x) = f(a X ). The relation (2)
transforms immediately into g(x + y) = g(x)g(y). Moreover, the function g,
the composition of f and of an exponential function, is continuous. By Theorem 1 there exists a b > 0 such that g(x) = b X • On the other hand, there
exists an s E lR. such that b = as, namely s = loga b. Hence
and, since any number y > 0 can be put in the form aX, we conclude that
f(y) = yS for all y > 0, qed.
Exercise. Find a direct proof of Theorem 4.
Theorem 5. Every real-valued function defined for x > 0 which is not identically zero, satisfies f(xy) = f(x) + f(y), and is monotone or continuous, is
a logarithmic function.
6 There are solutions of the equation f(x+y) = f(x)+ fey) which are neither continuous nor monotone. The relation in question certainly implies f(cx) = cf(x)
for c E 10, so that, if one considers lR as a vector space (of infinite dimension!)
over 10, every "linear form" on lR satisfies this condition. To construct one effectively, one would have to choose a "base" of lR over 10, requiring set theoretic
constructions calling on the theory of transfinite numbers or the axiom of choice.
These solutions are interesting only as curiosities.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
The reader will easily prove that in fact the relation
(6.5)
f(nx) = f(x)n for all x E lR. and all n E Z
is enough to characterise the exponential functions among the real continuous
or monotone functions.
Theorem 3. Every real-valued function defined on lR. which satisfies
f(x + y) = f(x) + f(y), and is monotone or continuous, is a linear function.
Put f(l) = a, and use (3) to show that f(x) = ax first for x E I'll, then
for x E Z, then for x E Q. The function being monotone or continuous, the
formula extends immediately to arbitrary real values of x.
Here, as above, the relation f(nx) = n.f(x) leads to the same result 6 . Note
further that this remains valid if f has complex values; but one must then
assume f continuous since the other possible hypothesis is then meaningless.
Theorem 4. Every real-valued function, defined for x > 0, which is not
identically zero, satisfies f(xy) = f(x)f(y), and is monotone or continuous,
is a power function.
Such a function in fact has positive values since f(x) = f(X 1 / 2 X 1 / 2 ) =
f(x 1 / 2 )2. It can never vanish, since f(c) = 0 would imply that f(x)
f(c)f(x/c) = 0 for all x > o.
Now let us choose an a> 0, not 1, and put g(x) = f(a X ). The relation (2)
transforms immediately into g(x + y) = g(x)g(y). Moreover, the function g,
the composition of f and of an exponential function, is continuous. By Theorem 1 there exists a b > 0 such that g(x) = b X • On the other hand, there
exists an s E lR. such that b = as, namely s = loga b. Hence
and, since any number y > 0 can be put in the form aX, we conclude that
f(y) = yS for all y > 0, qed.
Exercise. Find a direct proof of Theorem 4.
Theorem 5. Every real-valued function defined for x > 0 which is not identically zero, satisfies f(xy) = f(x) + f(y), and is monotone or continuous, is
a logarithmic function.
6 There are solutions of the equation f(x+y) = f(x)+ fey) which are neither continuous nor monotone. The relation in question certainly implies f(cx) = cf(x)
for c E 10, so that, if one considers lR as a vector space (of infinite dimension!)
over 10, every "linear form" on lR satisfies this condition. To construct one effectively, one would have to choose a "base" of lR over 10, requiring set theoretic
constructions calling on the theory of transfinite numbers or the axiom of choice.
These solutions are interesting only as curiosities.