An initial-value problem combines an ODE with the value of the unknown function at a starting point, for example

$$y'=f(t,y),\qquad y(t_0)=y_0.$$

Two separate questions matter: does at least one solution pass through $(t_0,y_0)$, and if so, is it unique?

Continuity of $f$ near the initial point is enough for local existence under standard hypotheses. Uniqueness requires stronger control over variation with $y$. A local Lipschitz condition requires some $L$ such that

$$|f(t,y_1)-f(t,y_2)|\le L|y_1-y_2|$$

near the initial point.

A convenient sufficient condition is that $\partial f/\partial y$ exists and is continuous there.

These hypotheses matter because an ODE formula alone need not determine one trajectory. Existence and uniqueness theorems state when an initial state defines a well-posed local evolution.