One normally defines an initial value problem by the following parameters:
Let $D \subseteq \mathbb{R} \times \mathbb{R}^{d}$ be an open set, $(t_{0},x_0)\in D$ and $f:D \rightarrow \mathbb{R}^d$ a continuous function. We consider the initial value problem given by $ (P) \begin{cases} x' = f(t,x) \\ x(t_{0}) = x_0 \end{cases} $
Why do we impose that $f$ is a continuous function?
In other words, what are the theoretical motivations for it (of course I'd say it is easier to work with this)? What happens if we remove this condition? (for instance I just discovered Carathéodory's theorem).
For initial value problems
$$\begin{cases} x' = f(t,x) \\ x(t_{0}) = x_0 \end{cases}$$
continuity of $f(t,x)$ implies the existence of a solution and continuity of $$\partial f/\partial x$$ implies uniqueness.