= Solution
The <function>
$$
y(t)=\begin{cases}
y_0-t,&0\leq t\leq y_0,\\
0,&y_0\leq t\leq T
\end{cases}
$$
is continuous, satisfies the equation away from $\widetilde t=y_0$, and has unequal one-sided <derivatives> there.
It is unique in the stated class. While a solution is positive its <derivative> is $-1$, so continuity forces it to follow $y_0-t$ until it first reaches zero at $t=y_0$. It cannot subsequently make a positive excursion: on any connected interval $(a,b)$ where $y>0$ and $y(a)=0$, integration of $y'=-1$ gives $y(t)=-(t-a)<0$. A negative excursion similarly contradicts $y'=1$. Hence the solution remains zero. This is the <sign-decay differential equation>.
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