A function f(x) is said to be continuous at x = a

if $\lim_{x\rightarrow a}f(x)$ = f(a)This definition implies three conditions necessary for continuity.

- Function is defined at x = a, that is f(a) exists.
- $\lim_{x\rightarrow a}f(x)$ exists
- $\lim_{x\rightarrow a}f(x)$ = f(a)

All the above three conditions are to be satisfied for a function to be continuous at x = a.

**The partial continuities of a function f(x) at a point x = a are defined as follows:**

A function f(x) is said to be continuous to the right of x = a if,

$\lim_{x\rightarrow a^{+}}f(x)$ = f(a)

and the function is said to be continuous to the left of x =a if,

$\lim_{x\rightarrow a^{-}}f(x)$ = f(a)

**Continuity on an interval**

A function is said to be continuous in an open interval (a, b), if it is continuous in every point x, such that a < x < b.

A function is said to be continuous in a closed interval [a, b], if the it continuous at every interior point and right continuous at x = a and left continuous at x = b.