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For an ordinary differential equation $\frac{dx}{dt}=f(t, x(t))$, under what conditions on $f$ there exists at least one solution on the interval $[t_0, T]$ passing through the point $(t_0, x_0)$?

Here $[t_0, T]$ and $(t_0, x_0)$ are prescribed by me. I want the solution to this equation should exists on $[t_0, T]$ and it should passes through $(t_0, x_0).$

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If $f$ is continuous and bounded on $[t_0, T] \times \mathbb R$, then the initial value problem

$x'=f(t,x(t))$ and $x(t_0)=x_0$

has a solution on $[t_0, T]$

(Existence - Theorem of Peano).

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    If $f$ is a function of $t,$ $x$ and $u$ then what will be the condition for the existence of a solution on $[t_0, T].$ which passes through $(t_0, x_0)?$ Here $u$ is also a function of $t.$ I don't know the nature of the function $u$ – 2017-02-13