How does the author of this probability book claim that a one-to-one function is invertible without showing it's onto? I know that the necessary conditions for a function to be invertible is that the function must be onto. I'm not sure how the author can just assume that the function $u(x)$ is onto. !enter image description here

The author is not up to modern standards in his treatment of functions:

(1) No mention is made of the domain or codomain of any function.

(2) The notation used muddles a function $f$ with its value $f(x)$ at a general point $x$.

It seems to be left to the reader to work out what the domain and range are for any function that is treated. When you have ascertained these for a one-to-one function, you can of course construct the inverse function from the range of the original function to its domain.