Find "$g(x)$" knowing that "$x=\int_{0}^{\infty} g(tx) dt$"???

Question

The entire question states what I am looking for. I'm looking for a function $g(x)$ in terms of $x$ which satisfies the condition that follows. This seems like it's related to "integral transformations," a topic I've seen before, but have never studied much. So, I don't even know where to begin trying to find $g(x)$. Thank you very much!

Answer 1

$\begin{array}\\ x &=\int_{0}^{\infty} g(tx) dt\\ &=\int_{0}^{\infty} g(y) dy/x \qquad t\,x = y, t = y/x, dt = dy/x\\ \text{so}\\ x^2 &=\int_{0}^{\infty} g(y) dy\\ \end{array} $

Looks like it holds for only one (or two) $x$.

Answer 2

If you really really want a function, try this one: The domain contains only the point $x=0$, and $g(0)=0$.