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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.2011-06-10

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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 whenever $a>0$ are solutions of this ODE.

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    @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.2011-06-10
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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.