Just as there is a distinction between the (countable) set of
integers and the (uncountable) set of real numbers, there is a
distinction between discrete groups (where an identification can be made
between group elements and the integers) and continuous groups (with an
uncountably infinite number of group elements).
is a
continous group, and is the basis of *calculus*, because it
supports the idea of *differentiation* using a suitable limiting
process such as

A Lie Group is a continuous group, which is also formally a
*differentiable manifold*. We could easily get swept down the
rabbit hole to ``real math'' at this point and explain that a
differentiable manifold is any space that is *locally* isomorphic
to a Euclidean (flat) space like
(a real space in three
orthogonal dimensions) wherein differentiation is well defined. This
means that a Lie group is generated by composing a large number of
*local* ``infinitesimal transformations'' into a finite
transformation. Continuous coordinate transformations in physics often
form Lie groups, in particular the set of all continous rotations of a
coordinate frame, SO(3).

All of this section so far, in fact, leads to this one conclusion.
Coordinate transformations of interest to us in physics in general, and
electrodynamics in particular, almost always end up being Lie groups
(with an associated *Lie algebra* for the abstract group
operations) generated from infinitesimal local transformations. The
continous groups are often extended by a (usually small/finite) set of
discrete transformations, such as inversion. Let's discuss this
further.