I have a delayed differential equation of the form $$ x'(t)=\beta-f\left(x\left(\alpha(t)\right)\right) $$ where $\alpha(t)=t_{j}$ for $t\in[t_{j},t_{j+1})$ for $j=0,1,...$, with $t_{0}=0$ and $t_{j+1}>t_{j}$ and $f$ is continuous. So one can write $x'(t)$ as \begin{eqnarray*} x'(t) & = & \beta-f\left(x\left(0\right)\right),\quad t\in[0,t_{1}),\\ x'(t) & = & \beta-f\left(x\left(t_1\right)\right),\quad t\in[t_{1},t_{2}),\\ x'(t) & = & \beta-f\left(x\left(t_2\right)\right),\quad t\in[t_{2},t_{3}),\\ & \vdots \end{eqnarray*} Then, given the continuity of $f$, starting from the first interval one can sequentially solve for $x$. That is given the initial condition $x(0)$ one can solve the ODE on the first interval, use it together with continuity to obtain $x(t_1)$ as the initial condition of the second equation and so on.
Question: Is this argument correct? If yes, is it also true that the existence and uniqueness of the solution follows from those of the solutions for all intervals?