of , we say that is continuous at if, for any , there exists such that for all , implies that . We say that is a continuous map if is continuous at every . A consequence of this definition is that is trivially continuous at any isolated point . This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between topological spaces (which includes metric spaces and in particular as special cases). This definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness. Definition. If and are topological spaces, we say that is continuous at if is a neighborhood of in for every neighborhood of in . We say that is a continuous map if is open in for every open in . (Here, refers to the preimage of under .) Uniform continuity Definition. If is a subset of the real numbers, we say a function is uniformly continuous on if, for any , there exists a such that for all , implies that . Explicitly, when a function is uniformly continuous on , the choice of needed to fulfill the definition must work for all of for a given . In contrast, when a function is continuous at every point (or said to be continuous on ), the choice of may depend on both and . In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point is meaningless. On a compact set, it is easily shown that all continuous functions are uniformly continuous. If is a bounded noncompact subset of , then there exists that is continuous but not uniformly continuous. As a simple example, consider defined by . By choosing points close to 0, we can always make for any single choice of , for a given . Absolute continuity Definition. Let be an interval on the real line. A function is said to be absolutely continuous on if for every positive number , there is a positive number such that whenever a finite sequence of pairwise disjoint sub-intervals of satisfies then Absolutely continuous functions are continuous: consider the case n = 1 in this definition. The