If all normal lines are parallel, then the surface is a plane. I have to prove the following statement:

If $S$ is a connected surface such that every normal line is parallel to a unit vector $a,$ then show that $S$ is a subset of a plane.

Now, for a point $p$ on $S,$ the plane $$T_p:=\{x\in\mathbb{R}:\langle x-p,a\rangle=0\}$$
Is the tangent plane at $p.$ It will be sufficient enough to show that every point $z\in S$ is in $T_p.$
I completely understand this geometrically, but have not been able to put it into words at all. I know that $S$ being connected is crucial...
A hint here would really go a long way. Thank you so much!

Consider the function $f:S\to\mathbb R$ given by $f(p)=p\cdot a$. Now for any point $p$ and any parametrised curve $C:[-1,1]\to S$ with $C(0)=p$ and $C'(0)=v\in T_p(S)$, the composition $\alpha=f\circ C$ satisfies
$$\alpha'(0)=f'(v)=a\cdot v=0$$
since the normal, a scaling of $a$, is perpendicular to $T_p(S)$. Thus $f$ is constant over $C$. Since $C$ was arbitrary and $S$ is connected, $f$ is constant over $S$ and $S$ is a plane.
