is metrizable corresponding (for a given homeomorphism) to the ordinary metric on this circle (either measured straight or along the circle). There is no metric which is an extension of the ordinary metric on Interval arithmetic Interval arithmetic extends to from . The result of an arithmetic operation on intervals is always an interval, except when the intervals with a binary operation contain incompatible values leading to an undefined result. In particular, we have, for every : irrespective of whether either interval includes and Calculus The tools of calculus can be used to analyze functions of . The definitions are motivated by the topology of this space. Neighbourhoods Let and . is a neighbourhood of x, if A contains an open interval B that contains . is a right-sided neighbourhood of , if there is a real number such that and contains the semi-open interval . is a left-sided neighbourhood of , if there is a real number such that and contains the semi-open interval . is a punctured neighbourhood (resp. a right-sided or a left-sided punctured neighbourhood) of , if and is a neighbourhood (resp. a right-sided or a left-sided neighbourhood) of . Limits Basic definitions of limits Let and . The limit of f&hairsp;(x) as x approaches p is L, denoted if and only if for every neighbourhood A of L, there is a punctured neighbourhood B of p, such that implies . The one-sided limit of f&hairsp;(x) as x approaches p from the right (left) is L, denoted if and only if for every neighbourhood A of L, there is a right-sided (left-sided) punctured neighbourhood B of p, such that implies It can be shown that if and only if both and . Comparison with limits in The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements, the first limit is as defined above, and the second limit is in the usual sense: is equivalent to is equivalent to is equivalent to is equivalent to is equivalent to is equivalent to Extended definition of limits Let . Then p is a limit point of A if and only if every neighbourhood of p includes a point such that Let , p a limit point of A. The limit of f&hairsp;(x) as x approaches p through A is L, if and only if for every neighbourhood B of