Existence at least one solution differential equation

Question

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 $(t_0, x_0).$

Answer 1

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).