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Given an arbitrary (not necessarily continuous) function of a topological space to itself, we associate a non-negative extended real number which we call the continuity entropy of the function. In the case where the space is compact and the function is continuous, the continuity entropy of the map is equal to the usual topological entropy of the map. We show that some of the standard properties of topological entropy hold for continuity entropy, but some do not. We show that for piecewise continuous piecewise monotone maps of the interval the continuity entropy agrees with the entropy dened in Horseshoes and entropy for piecewise continuous piecewise monotone maps by Michal Misiurewicz and Krystina Ziemian Finally, we show that if f is a continuous map of the interval to itself and g is any function of the interval to itself which agrees with f at all but countably many points, then the continuity entropies of f and g are equal.