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``The'' continuity axiom is an additional Axiom which must be added to those of Euclid's 
Elements in order to guarantee that two equal Circles of Radius 
intersect each
other if the separation of their centers is less than 
(Dunham 1990). The continuity axioms are the three of
Hilbert's Axioms which concern geometric equivalence.
Archimedes' Lemma is sometimes also known as ``the continuity axiom.''
See also Congruence Axioms, Hilbert's Axioms, Incidence Axioms, Ordering Axioms, Parallel Postulate
References
Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, p. 38, 1990.
Hilbert, D. The Foundations of Geometry. Chicago, IL: Open Court, 1980.
Iyanaga, S. and Kawada, Y. (Eds.). ``Hilbert's System of Axioms.'' §163B in
Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 544-545, 1980.