This is part of the derivation of the Ito integral. In particular extending the definition to more general functions.
I cannot understand why $g(.,\omega)$ is continuous for each $\omega$. $\psi$ does not seem to be continuous and neither is $h$. I cannot find any theorem that suggests all antiderivatives are continuous.
Hint:
First, note that $\psi$ is defined to be continuous (see top of proof). Then, if you define $\langle f, g\rangle := \int fg$, and derive a norm from it, it is not too hard to show that the integral of the product of a continuous and a bounded function (which is the case here) is continuous (use Cauchy-Schwarz). Note also that this is a "for each $\omega$" statement only at the point you underline.