Continuous homomorphisms between Lie groups are smooth

The graph ${\mathrm{\Gamma }}_F=\{(g,F(g)):g\in G\}$ is a subgroup of $G\times H$. It is closed: if $(g,h)$ is not on the graph, then $h\mathrm{\ne }F(g)$, and continuity of $F$ together with Hausdorffness of $H$ gives product neighborhoods of $(g,h)$ disjoint from the graph. By [A1], ${\mathrm{\Gamma }}_F$ is an embedded Lie subgroup.

The first projection $P:{\mathrm{\Gamma }}_F\to G$ is a smooth Lie-group homomorphism and a homeomorphism, with continuous inverse $g\mapsto (g,F(g))$. Manifold charts and [F1] therefore give $\dim {\mathrm{\Gamma }}_F=\dim G$.

We show that $d{P}_{(e,e)}$ is injective. If $X$ is in its kernel, [F3] gives $P({\exp }_{{\mathrm{\Gamma }}_F}(tX))={\exp }_G(t,d{P}_{(e,e)}X)=e$ for every $t$. The algebraic kernel of $P$ is the singleton $(e,e)$, so this one-parameter subgroup is constant; differentiating it at zero gives $X=0$. Equal dimensions from step 2.1 now make $d{P}_{(e,e)}$ an isomorphism.

Translation of the homomorphism identity makes $dP$ invertible everywhere. By [F2], $P$ has smooth local inverses around every point. Since the set-theoretic inverse is unique, these local inverses agree with the global continuous inverse $P^{-1}$, so $P^{-1}$ is smooth.

Let $Q:{\mathrm{\Gamma }}_F\to H$ be the second projection. Then $F=Q\circ P^{-1}$ is smooth. Zero-dimensional and disconnected groups are included; no injectivity or surjectivity of $F$ is assumed. The exponential argument in step 3.1 is the needed correction to the scaffold: a smooth homeomorphism between equal-dimensional manifolds need not have invertible differential. Countable choice is used through [A1] and [F3].