Prove or disprove ; 

Let $S $ be a surface in $ R^3$.
 $S $ is a plane iff every point of $S $ is planar point.

"All points of plane are planar points" is trivial.
But,... the converse is also really true?

The definition of planar point ; $p $ is called a planar point of $S $
iff the two principal curvatures vanish.