Dynamics as the preservation of a constant commutator

Some properties involving two operators with a constant commutator are derived. They include a definition of derivatives of operator functions, their conjugate spaces, and the associated translation generators. The cases of real functions, quantum coordinate and momentum operators, time and Liouville operator, and quantum time and energy operators, are analyzed within this formalism. This procedure allows the elucidation of the properties of time in classical and quantum mechanics.