In convergence of adaptive algorithm, why we assume $|H(\theta, x)| \leq C_1(1 + |x|^{q_1})$?

Question

In the book Adaptive Algorithms and Stochastic Approximations Part 2, P.216, the author assume $|H(\theta, x)| \leq C_1(1 + |x|^{q_1})$.

I am now reading a paper that applies the concepts from this book. While I grasp the practical aspects, the theoretical underpinnings elude me. I don't have any background on ODE, or measure theory, so I cannot figure out why the author assumes such a condition. Would you mind providing some keywords or some theory for me to self-study?

I have googled Lipschitz continuous, can it apply to single function? e.g. function in (A.3.):

Thank you!