Discontinuous functions

A function f is said to be discontinuous at a point a of its domain D if it is not continuous threat. The point a is then called a point of discontinuity of the function. The discontinuity may arise due to any of the following situations:

a)  or  of both may not exist.

b)  as well as  may exist, but are unequal.

c)  as well as  both exist, but either of the two or both may not be equal to f(a).

We classify the points of discontinuity according to various situations discussed above.

1. Removable discontinuity: A function f is said to have removable discontinuity at x = a if  but their common value is not equal to f(a). Such a discontinuity can be removed by assigning a suitable value to the function f at x = a.

2. Discontinuity of the first kind: A function f is said to have a discontinuity of the first kind at x = a if   and  both exist but are not equal. f is said to have a discontinuity of the first kind from the left at x = a if  exists but not equal to f(a). Discontinuity of the first kind from the right is similarly defined.

3. Discontinuity of second kind: A function f is said to have a discontinuity of the second kind at x = a if neither  nor  exists.

A function f is said to have discontinuity of the second kind from the left at x = a if  does not exist. Similarly, if  does not exist, then f is said to have discontinuity of the second kind from the right at x = a.