Problem 99 **

Find all continuous functions f : (1, \infty) \to \mathbb R which satisfy f(xy) = xf(y) + yf(x) for all x, y > 1.

[originally posted by Mladenov on TSR]

We can divide by xy since xy>1
\frac{f(xy)}{xy}=\frac{f(y)}{y}+\frac{f(x)}{x}
Let g(x)=\frac{f(x)}{x}
\therefore g(xy)=g(y)+g(x)
This is a well know functional equation with general (continuous) solution: g(x)=C\ln x, \ C \in \Bbb R
Therefore, f(x) must have general solution f(x)=Cx\ln x, \ C \in \Bbb R