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(or point of discontinuity), a value of the argument at which the continuity of a function is violated. In the simplest case, continuity is violated at some point a in the sense that the right and left limits
exist, but at least one of them differs from f(a). When this occurs, a is called a jump discontinuity of f, or the discontinuity of the first kind.
If f(a + 0) = f(a – 0), the discontinuity is said to be removable, since f(x) becomes continuous at a if we set f(a) = f(a + 0) = f(a – 0). For example, the point a = 0 is a removable discontinuity of the function
since f is continuous at 0, if we set f(0) = 1. If, however, the jump δ = f(a + 0) – f(a – 0) of the function f(x) at the point is nonzero, then a is a discontinuity for any definition of the value of f(a). An example of such a discontinuity is the point a = 0 for the function f(x)= arc tan 1/x. In this case, the function may not be defined at the point a. The jump discontinuity is called regular if the condition f(a) = ½[f(a – 0) + f(a + 0)] is satisfied. If either of the one-side limits does not exist, then the point a is called the discontinuity point of the second kind. Examples are the point a = 2 for the function f(x) = 1/(x – 2) and the point a = 0 for f(x) = sin 1/x.