The function \(f\) is such that for any positive \(x\) and \(y\) the equality \(f (xy) = f (x) + f (y)\) holds. Find \(f (2007)\) if \(f (1/2007) = 1\).
Find all functions \(f (x)\) defined for all positive \(x\), taking positive values and satisfying the equality \(f (x^y) = f (x)^f (y)\) for any positive \(x\) and \(y\).
The function \(f (x)\) is defined and satisfies the relationship \((x-1) f((x=1)/(x-1)) - f (x) = x\) for all \(x \neq 1\). Find all such functions.
Find all the functions \(f\colon \mathbb {R} \rightarrow \mathbb {R}\) which satisfy the inequality \(f (x + y) + f (y + z) + f (z + x) \geq 3f (x + 2y + 3z)\) for all \(x, y, z\).
Is there a bounded function \(f\colon \mathbb{R} \rightarrow \mathbb{R}\) such that \(f (1)> 0\) and \(f (x)\) satisfies the inequality \(f^2 (x + y) \geq f^2 (x) + 2f (xy) + f^2 (y)\) for all \(x, y \in \mathbb{R}\)?
For which \(\alpha\) does there exist a function \(f\colon \mathbb{R} \rightarrow \mathbb{R}\) that is not a constant, such that \(f (\alpha (x + y)) = f (x) + f (y)\)?
On a function \(f (x)\) defined on the whole line of real numbers, it is known that for any \(a > 1\) the function \(f (x)\) + \(f (ax)\) is continuous on the whole line. Prove that \(f (x)\) is also continuous on the whole line.
Does there exist a function \(f (x)\) defined for all \(x \in \mathbb{R}\) and for all \(x, y \in \mathbb{R}\) satisfying the inequality \(| f (x + y) + \sin x + \sin y | < 2\)?
On a function \(f (x)\), defined on the entire real line, it is known that for any \(a>1\) the function \(f (x) + f (ax)\) is continuous on the whole line. Prove that \(f (x)\) is also continuous on the whole line.
Find all functions \(f (x)\) defined for all real values of \(x\) and satisfying the equation \(2f (x) + f (1 - x) = x^2\).