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I' ve found in a book without a proof the following theorem: Let $A=I\times \mathbb{R}$, where I=]a,b[ is an interval with $-\infty\leq a<b\leq +\infty$, be a set in which the function f(x,y): $(x,y) \subset A \rightarrow \mathbb{R} $ is defined, continous and locally lipschitzian in y uniformly in x . Let \begin{cases} y'(x)=f(x,y) \\ y(x_0)=y_0 \end{cases} be a Cauchy problem with $x_0\in I$. If $\tilde{y}$ is a not exctendable unique solution of the problem defined in a maximal interval $]a_{m},b_m[$ then we have that:
(i)$b=b_m$ or $b_m<b $ and $ \lim_{x \to b_m^{-}} |\tilde{y}(x)| = +\infty$
(ii)$a=a_m$ or $a_m>a $ and $ \lim_{x \to a_m^{+}} |\tilde{y}(x)| = +\infty$
Now I ask where can I find the proof of this theorem and the possible usability in finding the maximal interval of existency of the solution of a Cauchy problem.

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A more general statement is that for any ODE IVP $$y'(x)=f(x,y(x)),~~y(x_0)=y_0$$ where $f$ has a (open) domain $$D\subset \Bbb R\times \Bbb R^n,~~ $(x_0,y_0)\in D,$$ and $f$ continuous and locally Lipschitz in $y$ on $D$, ... , then the (unique) maximal solution leaves any compact set inside of $D$.

The proof is essentially that any solution that ends in a compact set has a limit point, and can thus be continued by the local solution of the IVP starting at the limit point.

In your situation, take the compact sets $[a+\delta,b-\delta]\times[-N,N]\subset D$.

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