$t$-continuity of Itô Integral

Question

I'm reading Øksendal's SDEs book and in Theorem 3.2.5 he proves that the Itô integral $$ \int_0^t f(s,\omega) dB_s(\omega)$$ has a version such that it is $t$-continous. In order to do this he picks elementary functions $\phi_n$, such that $$ E[\int_0^T (f-\phi_n)^2 dt]\rightarrow 0$$ as $n$ tends to infinity. Then he defines $I_n(t,\omega)=\int_0^t \phi_n(s,\omega)dB_s(\omega)$ and states that $I(\cdot,\omega)$ is continous for all $n$. This is crucial to apply Doob's martingale inequality later and gives the statement without proof, but this doesn't seem trivial to me. My question is, why should be the Itô integral $I_n(t,\omega)$ be continous at all? Aren't there going to be jumps as we increase $t$ and more addends appear in the integral?