= Solution
{c}
Suppose $DF_x$ were invertible at some $x\in\Omega$. The <inverse function theorem> supplies a neighborhood $V$ of $x$ for which $F(V)$ is a nonempty open subset of $\mathbb R^2$. But $F(V)\subset M$, and a one-dimensional <embedded submanifold> of $\mathbb R^2$ has empty interior. To see this last assertion, a <slice chart for an embedded submanifold> takes $M$ locally into a coordinate line; a coordinate line cannot contain an open plane neighborhood, and a chart preserves openness. This is a contradiction.
Thus $DF_x$ is singular everywhere. Part (c)(i) strengthens singularity to $DF_x=0$ everywhere. By (b) and <connectedness> of $\Omega$,
$$
\boxed{F\text{ is constant}.}
$$
The special form of the <Jacobian matrix> is essential: an arbitrary differentiable map can have nonconstant image in a curve with derivative of rank one.
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