This paper extends the idea of the initialization function to the more general concept of a continuation function. The paper sets forth definitions for operator self-consistency which are then applied to test three operators, the ordinary Riemann integral, the time-varying initialized Riemann-Liouville fractional integral, and finally the Caputo derivative. Self-consistency was found for the first two cases. The Caputo fractional derivative operator was found to be self-inconsistent based on possible continuation functions derived from the Laplace transform of the derivative. A theoretical continuation function was derived which does make the derivative self-consistent, but requires a time-varying initialization function negating a primary attraction of this derivative.

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