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Let $f$ be a continuous real function defined in a subset of the real line.The standard definition of continuity at a point $x$ allow us to correlate anygiven epsilon with a (possibly depending of $x$) delta value. This pairing isknown as the epsilon--delta relation of $f$. In this work, we demonstrate theexistence of a privileged choice of delta in the sense that it is continuous,invertible, maximal and it is the solution of a simple functional equation. Wealso introduce an algorithm that can be used to numerically calculate this mapin polylogarithm time, proving the computability of the epsilon--deltarelation. Finally, some examples are analyzed in order to showcase the accuracyand effectiveness of these methods, even when the explicit formula for theaforementioned privileged function is unknown due to the lack of analyticaltools for solving the functional equation.

A Polylogarithm Solution to the Epsilon--Delta Problem