find all functions f:Q—>R for which f(x+y)=f(x)+f(y),∀x, y belongs to Q
Answers
Cauchy’s equation
Let us begin by restating and solving Cauchy’s functional equation. Let f :
R → R be a continuous function satisfying
f(x + y) = f(x) + f(y) (2.1)
for all real x and y. We show that there exists a real number a such that
f(x) = a x for all x ∈ R.
It is straightforward to show by mathematical induction that (2.1) implies
f(x1 + x2 + ··· + xn) = f(x1) + f(x2) + ··· + f(xn) (2.2)
for all x1, ..., xn ∈ R. A special case of this is found by setting x1 = ··· =
xn = x, say. Then (2.2) becomes
f(n x) = n f(x) (2.3)
for all positive integers n and for all real x. Let x = (m/n)t, where m and n
are positive integers. Then n x = m t. So
f(n x) = f(m t)
n f(x) = m f(t)
n f m
n
t
= m f(t).
But this can be written as
f
m
n
t
= m
n
f(t) (2.4)
for all t ∈ R. Thus we have proved that