Corner compatibility for constant-speed transport ($f^{(j)}(0)=(-a{)}^j{u}_0^{(j)}(0)$)

For the transport equation $u_t+a{u}_x=0$ on the quadrant with $a>0$, smooth initial and inflow data glue across $x=at$ to order $m$ precisely when $f^{(j)}(0)=(-a{)}^j{u}_0^{(j)}(0)$ for $0\le j\le m$. Matching all orders yields a smooth solution; matching orders zero and one yields a classical solution up to the two boundaries.