We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. Show that we can define a sequence of step functions $\psi_N : [0, T] \rightarrow \mathbb{R}^d$ such that $\psi_N(t) = \phi_N(t)$ for $t \in \{n\Delta t \mid n \in \{0, \cdots, N\}\}$ and such that:
$$\forall N \in \mathbb{N}^*, \phi_N(t) = y_{\text{init}} + \int_0^t F(\psi_N(s))\, ds \text{ for all } t \in [0, T]$$
We will specify the expression of the functions $\psi_N$.