I'd like to find some theorem, lemma, proposition or intuitive explanation to this.
Prove that the function $f:\mathbb{R}\times \mathbb{R} \to \mathbb{R}$ defined as
$$f(t,x)=\begin{cases}\frac{tx}{t^2+x^2}\ \ \ if \ \ \ (t,x)\neq(0,0) \\ 0 \ \qquad if \ \ \ (t,x)=(0,0).\end{cases}$$
is not continuous on the origin, but the IVP $$\begin{cases}\dot{x}=f(t,x) \\ x(t_0)=x_0\end{cases}$$
Has a solution for all $(t_0,x_0)\in\mathbb{R^2}$.
Every theorem that I know uses continuity of $f(t,x)$ to say that there exists a solution to an IVP.
It's necessary to solve this problem? Or we can find some theorem to say that there exists a solution to this IVP?
Caratheodory's existence theorem gives to answer to your question. It only needs continuity of $f$ in $x$ for fixed $t$.