Continuity at a Point

Continuous Meaning: We say a function f (x) is continuous at a point x = a it means at a point (a, f (a)). The graph of the function has no holes or graphs. That is, its graph is unbroken at point (a, f (a)).

The continuity at x = a₂ if  does not exist

Continuity at a point: A function f (x) is said to be continuous at a point x = a of its domain if 

Thus, (f (x) is continuous at x = a)

⇒ If f (x) is not continuous at a point x = a, then it is said to be discontinuous at x = a.
⇒ If , then the discontinuity is known as the removable discontinuity because f(x) can be made continuous by re-defining it at point x = a in such a way that .

⇒ If , then f (x) is said to have a discontinuity of first kind.

⇒ A function f (x) is said to have a discontinuity of the second kind at x = a if  or  or both do not exist.

Continuity on an open interval:  A function f (x) is said to be continuous on an open interval (a, b) if it is continuous at every point on the interval (a, b).

Continuity on a closed interval:  A function f (x) is said to be continuous on a closed interval {a, b} if

1. f is continuous on the open interval (a, b)

2. 

and,

3. 

In order words, f (x) is continuous on [a, b] if it is continuous on (a, b) and it is continuous at a form the right and at b form the left.