# Does differential equation always has solution if vector field is only continuous? [duplicate]

Let $V$ be a $continuous$ vector-field defined on a domain $U$ in $R^n$ . Let $x_0$ be a point in $U$. Is it true that the differential equation $\frac{dx}{dt}$ = $V(x)$ admits a solution $\phi(t)$ satisfying initial condition $\phi(0)$=$x_0$ defined on some open interval containing $0$.

By Arnold's Book Existence theorem I know that unique solution exists in case of smooth vector field.But here vector field is only $continuous$.Can we have counterexample in this case?

## marked as duplicate by Hans Lundmark, user147263, timur, Hagen von Eitzen, Lost1Oct 9 '14 at 18:53

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• Funny. This was asked yesterday too. – Hans Lundmark Oct 9 '14 at 16:41

## 1 Answer

Yes.It is called Peano's existence theorem.

• does Peano's existence extend to higher dimensions? – Kayoken Oct 9 '14 at 16:30
• Yes. However, you loose uniqueness (in all dimensions). – Julián Aguirre Oct 9 '14 at 16:31
• Oh i see..i wonder why in arnold's book smooth vector fields are only considered – Kayoken Oct 9 '14 at 16:34
• I guess it is because in most physical applications you want uniqueness of solution. – Julián Aguirre Oct 9 '14 at 16:46