= Solution
The left-hand value at the junction is
$$
y(\mu-)=1-e^{-\mu}-e^{-\mu}\sinh\mu=\frac12-e^{-\mu}+\frac12e^{-2\mu}.
$$
The right-hand value is
$$
y(\mu+)=(\cosh\mu-1)e^{-\mu}=\frac12+\frac12e^{-2\mu}-e^{-\mu}.
$$
Thus the piecewise <Green function> solution is <continuous>, with
$$
\boxed{y(\mu)=\frac12(1+e^{-2\mu}-2e^{-\mu})=\frac12(1-e^{-\mu})^2.}
$$
Its <derivative> is continuous as well: both one-sided values equal $-\tfrac12(1-e^{-\mu})^2$. Only the second <derivative> jumps, consistently with the step in the forcing and the absence of a <Dirac delta> source.
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