conditional statement saying "if , then "). Equivalently, is said to be uniformly continuous if . Here quantifications (, , , and ) are used. Alternatively, is said to be uniformly continuous if there is a function of all positive real numbers , representing the maximum positive real number, such that for every if then . is a monotonically non-decreasing function. Definition of (ordinary) continuity is called continuous if for every real number there exists a real number such that for every with , we have . The set is a neighbourhood of . Thus, (ordinary) continuity is a local property of the function at the point . Equivalently, a function is said to be continuous if . Alternatively, a function is said to be continuous if there is a function of all positive real numbers and , representing the maximum positive real number, such that at each if satisfies then . At every , is a monotonically non-decreasing function. Local continuity versus global uniform continuity In the definitions, the difference between uniform continuity and continuity is that, in uniform continuity there is a globally applicable (the size of a neighbourhood in over which values of the metric for function values in are less than ) that depends on only while in continuity there is a locally applicable that depends on the both and . Continuity is a local property of a function — that is, a function is continuous, or not, at a particular point of the function domain , and this can be determined by looking at only the values of the function in an arbitrarily small neighbourhood of that point. When we speak of a function being continuous on an interval, we mean that the function is continuous at every point of the interval. In contrast, uniform continuity is a global property of , in the sense that the standard definition of uniform continuity refers to every point of . On the other hand, it is possible to give a definition that is local in terms of the natural extension (the characteristics of which at nonstandard points are determined by the global properties of ), although it is not possible to give a local definition of uniform continuity for an arbitrary hyperreal-valued function, see below. A mathematical definition that a function is continuous on an interval and a definition that is uniformly continuous on are structurally similar