Say we had an IVP where $f(t,x)$ is a continuous function. \begin{align*} x' &= f(t,x) \tag{1}\\ x(t_0) &= x_0, \end{align*} where an integral equation is defined as a solution on interval $I$ with $t_0 \in I$ as \begin{align*} x(t) &= x_0 + \int_{t_0}^t f(s,x(s))ds \tag{2} \end{align*} where $x(t)$ is a solution to the integral equation.
I wouldn't think eqn (2) would need a solution to be continuous as integrals don't need to be continuous to be integrated. But what about for eqn (1)? I am a little confused on this as it wasn't clear in lecture. Where would it need to be continuous?