Find a function continuous on $[0,2]$ differentiable on $(0,1) ,(1,2)$

Question

I need to find a function $\Psi:[0,2] \to \mathbb{R}$ which is

1. Continuous on $[0,2]$.
2. Differentiable on $(0,1),(1,2)$.
3. Satisfies initial condition $\Psi (t=0)=1$.
4. Satsifies the ODE $ d_t \Psi= -\Psi + F$ on $(0,1)$ and $(1,2)$, where $F$ is the function $F=0$ if $t\leq 1$, $F=1$ otherwise.

I can find a function that satisfies three of the four requirements but am struggling to find one that satisfies all four. For example $\Psi (t) =e^{-t} +F(t)$ satisfies all conditions except continuity. Any advice would be much appreciated.