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I'm wondering if the continuity of $f(t,y)$ is necessary for the existence and uniqueness of the solution to $dy/dt=f(t,y(t))$.

I think the existence and uniqueness only require $f(t,y)$ has a continuous second partial differential.

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You should say what you have edited in your question such that the answer posted do not appear as being wrong. And assuming continuous second partial differentiation is a stronger condition than Lipschitz continuity. – Beni Bogosel Jun 10 '11 at 8:18

If $y$ lives in a Banach space, $f$ need to be continuous in $t$ and locally Lipschitz continuous with respect to $y$ (for example, $C^1$ ) to guarantee the existence and uniqueness, also known as Cauchy-Lipschitz theorem.

If we only suppose that $f$ is continued, the uniqueness is not guaranteed, but we still have existence (Cauchy-Peano-Arzelà theorem). Because we use the local Lipschitz continuity to apply fixed-point theorem for Banach spaces in the proof of Cauchy-Lipschitz theorem.

Here is a counter-example if we do not suppose Lipschitz continuity: $$\begin{cases} y'(t) = 2 \sqrt{y(t)} \\ y(0) = 0 \end{cases}$$

All functions $y_a(t) = (t-a)^2$ for $t>a$ and $y_a(t) = 0$ for $t<a$ whenever $a>0$ are solutions of this ODE.

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I have no problem with locally Lipschitz. Can you show me why the uniqueness require the continuity of f? – user11869 Jun 9 '11 at 21:11
I just gave a counter-example, is that answer your question? – HYL Jun 9 '11 at 21:31
My question is about existence instead of uniqueness. So can you give me an example where no solution exists because $f(t,y)$ is not continuous. – user11869 Jun 9 '11 at 21:36
There are so many functions which are not continuous (and ugly)! You can pick one of them and check by yourself if some solution exists. – HYL Jun 9 '11 at 21:43
@user11869: The theorem of Peano proves that continuity of $f$ implies existence of solutions. I think there exist examples where $f$ is not continuous and we can find solutions of the equation. – Beni Bogosel Jun 10 '11 at 8:01

About the existence of solutions for some discontinuous functions $f$, you can check out the Caratheodory theorem, which proves the existence of solutions for a larger class of functions than the class of continuous functions.

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