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) \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) or \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) of both may not exist.
b) \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) as well as \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) may exist, but are unequal.
c) \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) as well as \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) 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.
Removable discontinuity: A function f is said to have removable discontinuity at x = a if \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) 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.
Discontinuity of the first kind: A function f is said to have a discontinuity of the first kind at x = a if \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) and \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) both exist but are not equal. f is said to have a discontinuity of the first kind from the left at x = a if \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) exists but not equal to f(a). Discontinuity of the first kind from the right is similarly defined.
Discontinuity of second kind: A function f is said to have a discontinuity of the second kind at x = a if neither \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) nor \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) exists.
A function f is said to have discontinuity of the second kind from the left at x = a if \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,f\left( x \right)\) does not exist. Similarly, if \(\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)\) does not exist, then f is said to have discontinuity of the second kind from the right at x = a.