Below are snippets of Chapter 3 of Huber's "Robust Statistics. They are needed to derive the asymptotic normality of M-estimator. However, I am concerning the step going from (2.17) to (2.18) i.e. how did he get $$ P\{T_n^{*}<t)=P\Big\{\sum \psi(x_i;t)\leq 0\Big\} $$ The author mentioned something about the continuity of $t$ and I can sort of see it graphically (if the $\sum \psi(x_i;t)$ is a continuous random variable for example) but I cannot use it to fill in the missing detail formally.
