Lens optics and the continuity problems of the ABCD matrix

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.