Definition of a solution of a first order ODE

[jacksonjs20]

Given an open connected subset $D$ of the $(t,x)$ plane and a function $f\in C(D,\mathbb{R})$, we say $\varphi\in C^1(\text{proj}_1D,\mathbb{R})$ is a solution of the first order differential equation $x'=f(t,x)$ if and only if $$\forall t\in \text{proj}_1D,\quad (t,\varphi(t))\in D$$ and
$$\forall t\in I, \quad \varphi'(t)=f(t,\varphi(t)) .$$

$\textbf{Question}$: Is there a way to alter this definition so that the first condition after the 'iff' is automatically satisfied?

Thanks in advance for any help.

 

[gtoguy97]

One possible approach is to replace the first condition with a different one: we say \varphi\in C^1(\text{proj}_1D,\mathbb{R}) is a solution of the first order differential equation x'=f(t,x) if and only if for all t\in I, there exists some point (t,\varphi(t))\in D such that \varphi'(t)=f(t,\varphi(t)). This ensures that the solution does not exist outside of the domain D.