The following is an exercise from Extreme Values, Regular Variation and Point Processes.
We say a function $f \colon (0,\infty) \to (0,\infty)$ is regularly varying with index $\rho$ if for all $x>0$ $\lim_{t \to \infty} \frac{f(tx)}{f(t)} = x^\rho$.
In the $a(x) \in RV_\rho$(regularly varying with index $\rho$) , $\rho \neq 0$. If $N_n, n\geq 1$, is non-negative random variable with $\frac{N_n}{n} \to N$ in probability, then $\frac{a(N_n)}{a(n)} \to N^\rho$ in probability.
My attempt:
There is a proposition in that book that says we may assume $a$ is a.e. continuous(absolutely continuous here means having a density, and we assume all functions are locally integrable which gives us a.e. continuity) 
We know that $\frac{a(Nn)}{a(n)} \to N^\rho$ almost surely, so I'd like $\frac{a(Nn)}{a(N_n)} \to 1$ in probability, but we cannot apply continuous mapping theorem with $a$ being continuous a.e. since the $Nn$ may have positive measure on the point of discontinuity, and even if we assume $a$ is continuous everywhere, I do not see how the it follows from $\frac{N_n}{n} \to N$ in probability.
Thank you.