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) $$\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f\left( x \right)$$ 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.