  Paraxial lens optics is discussed to study the continuity properties of the
$ABCD$ beam transfer matrix. The two-by-two matrix for the one-lens camera-like
system can be converted to an equi-diagonal form by a scale transformation,
leaving the off-diagonal elements invariant. It is shown that the matrix
remains continuous during the focusing process, but this transition is not
analytic. However, its first derivative is still continuous, which leads to the
concept of "tangential continuity." It is then shown that this tangential
continuity is applicable to $ABCD$ matrices pertinent to periodic optical
systems, where the equi-diagonalization is achieved by a similarity
transformation using rotations. It is also noted that both the scale
transformations and the rotations can be unified within the framework of
Hermitian transformations.
