Functional equations over R
Problem 1. Let f : R → R be a function such that f (x + y) = f (x) + f (y) for any x, y ∈ R. Prove that
a) f (nx) = nf (x) for any n ∈ Z, and any x ∈ R.
b) f (n)
= nf (1) for any n ∈ Z.
1
1
c) f
= f (1), for any q ∈ Z∗ .
q
q
p
p
d) f
= f (1), for any p ∈ Z and any q ∈ Z∗ .
q
q
e) If f is continuous in a point, then f (x) = xf (1), for any x ∈ R.
f ) If f is monotonic, then f (x) = xf (1), for any x ∈ R.
g) If f is also multiplicative, i.e. f (xy) = f (x)f (y), then f (x) = x, for any x ∈ R, or f (x) = 0 for any
x ∈ R.
Problem 2. Find the continuous functions f : D → R satisfying
a) f (x + y) = f (x)f (y), for any x, y ∈ D = R.
b) f (xy) = f (x) + f (y), for any x, y ∈ D = (0, ∞).
c) f (xy) = f (x)f (y), for any x, y ∈ D = (0, ∞).
p
Problem 3. Find all continuous real-valued functions of one real variable f (x) such that f ( x2 + y 2 ) =
f (x) + f (y), for all real x and y.
p
Problem 4. Find all continuous real-valued functions f (x) of one real variable such that f ( x2 + y 2 ) =
f (x)f (y) for all real x and y.
Problem 5. Determine the continuous function f : R → R such that f (x + y) = f (x) + f (y) − 2f (x)f (y).
Problem 6. Determine the continuous function f : R → R such that f (x + y) = f (x) + f (y) + 2xy − 1.
Problem 7. Let a, b ∈ R, |a| 6= 1. Determine the continuous function f : R → R such that f (ax + b) =
f (x).
Problem 8. Is there any continuous function f : R → R, satisfying the condition f (f (x)) = −x?
Problem 9. For which real numbers k does there exist a continuous real-valued function f (x) such that
f (f (x)) = kx9 for all real x?
x
Problem 10. Solve the functional equation f (x) = f
, x 6= 1, subject to the condition that f is
1−x
continuous at x = 0.
Problem 11. Find all real valued functions such that f (x) + 2xf (1 − x) = 2x + 1.
Problem 12. Find all real valued functions f (x) defined for all x 6= 0, 1 which satisfy the functional
equation f (x) + f (1 − x−1 ) = 1 + x.
Problem 13. Find the functions f : (0, π2 ) → R such that
f (x)
f (x) = x tan
.
tan x
1
Problem 14. Let a be a fixed real number. Find the functions f : R → R such that f (0) = , and
2
satisfying for any x, y ∈ R
f (x + y) = f (x)f (a − y) + f (y)f (a − x)
Problem 15. Find all the functions f : (0, ∞) → (0, ∞) satisfying the conditions
i) f (xf (y)) = yf (x), for all x, y > 0
ii) lim f (x) = 0
x→∞
Problem 16. Let f : R → (0, ∞) be a function with the properties
1. f (x) = 0 if and only if x = 0
2. f (x + y) ≤ f (x) + f (y) for any x, y
3. f (xy) = f (x)f (y) for any x, y
4. f monotonic on (0, ∞)
5. f (2) = 2
Then f (x) = |x|.
2
Functional inequations
Problem 17. Determine all the functions f : N → R satisfying for all k, m, n ∈ N the inequality f (kn) +
f (km) − f (k)f (mn) ≥ 1.
Problem 18. If f : R → R is an increasing bijection, find the functions g : R → R satisfying (f ◦ g)(x) ≤
x ≤ (g ◦ f )(x).
Problem 19. Determine all the functions f : R → R satisfying f (x − y) − xf (y) ≤ 1 − x, for all x, y ∈ R.
Functional equations over Z
Problem 20. Prove that f : Z → Z, f (n) = 1 − n is the only integer-valued function defined on the
integers that satisfies the following conditions.
(i) f (f (n)) = n, for all integers n;
(ii) f (f (n + 2) + 2) = n for all integers n;
(iii) f (0) = 1.
Problem 21. A function f : N → C is called completely multiplicative if f (1) = 1 and f (mn) = f (m)f (n)
for all positive integers m and n. Find all completely multiplicative functions f with the property that the
function F (n) = f (1) + f (2) + .. + f (n) is also completely multiplicative.
Problem 22. Determine, with proof, all the functions f : Z → Z which satisfy f (x + f (y)) = f (x) + y
for all integers x, y.