Before defining Ito integral for general $L_2$ processes usually textbooks first construct Ito integral for simple processes. A process $A_s$ is simple if there exist times $0 < t_1 < t_2 < ... < t_N$ such that $A_t = Y_j, \hspace{1cm} t_j \leq t \leq t_{j+1}$.
Then Ito integral is defined as $Z_{t_j} = \sum_{i = 0}^{j-1}Y_i(B_{t_{i+1}}- B_{t_i})$
This definion gives the process $Z$ defined at partition $0 < t_1 < t_2 < ... < t_N$. What I don't understand is how then can say that the function $t \to Z(t)$ is almost sure continuous function.
The definition they give doesn't allow to know the value of process $Z$ at points other than partition points. Even if we assume that $Z_t = Z_{t_{j}}$ for $t_j \leq t < t_{j+1}$ the process $Z$ would look like step function which is not continuous
Could someone explain how they know that $Z$ has continuous paths?
Lawler's stochastic calculus book, page 86