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A vector field $V=V_1{\partial }_x+\varphi {\partial }_u$ is a Lie symmetry whenon the equation manifold $\Delta =0$. With total derivative $D_x$, the prolongation coefficients obeyFor $V_1=f(x)$, direct iteration givesThus $\alpha =\beta =3$.

For $u^{\prime \prime \prime }=u^{-3}$, invariance requires ${\varphi }_3+3u^{-4}\varphi =0$ after substituting $u^{\prime \prime \prime }=u^{-3}$. Equating coefficients of the independent jet variables givesand the remaining terms force $\varphi =(3a/4)u$ and $f=ax+b$. Hence the Lie algebra is spanned by

Fix $z_0\notin \Gamma$ and put $\delta =\mathrm{dist}(z_0,\Gamma )>0$. If $|z-z_0|<\delta /2$, thenuniformly for $w\in \Gamma$. Multiplication by the bounded continuous function $f$ and integration therefore prove continuity of $F$ at $z_0$.

Take $\Gamma$ to be the positively oriented unit circle and $f\equiv 1$. By the Cauchy integral formula, $F(z)=1$ inside the circle and $F(z)=0$ outside it, so no continuous extension across $\Gamma$ exists.

For the final assertion, cover $K$ by finitely many sufficiently small closed axis-parallel squares whose slightly enlarged squares lie in $\Omega$, choosing the grid so that no boundary meets $K$. Apply the square Cauchy formula to $f$ on every selected square and add the results. Integrals over shared edges cancel with opposite orientations. The remaining finitely many oriented boundary polygons can be split into contours ${\Gamma }_j\subset \Omega \setminus K$, and their sum isfor every $z\in K$.