= Solution
At an ordinary point or a massless join, the string is continuous, so
$$
\boxed{\Psi(x_0^-,t)=\Psi(x_0^+,t)}.
$$
With no concentrated transverse <force> or point <mass>, transverse <force> balance requires
$$
\boxed{T_1\Psi_x(x_0^-,t)=T_2\Psi_x(x_0^+,t)}.
$$
For the common <tension> in this problem, the slope $\partial\Psi/\partial x$ is therefore continuous as well. A jump in <linear mass density> changes <wave speed>, but introduces no finite inertial <force> at a zero-length join. A point <mass>, external support <force>, or unequal tensions would require modified <boundary conditions>.
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