Continuity of a Function

A moving physical object cannot vanish at some point and reappear someplace else to continue its motion. Thus, we perceive the path of a moving object as a single, unbroken curve without gaps, jumps or holes. Such curves can be described as continuous. In this tutorial we shall discuss this intuitive idea mathematically and develop some properties of continuous curves.

Definition of Continuous Function:
The function f is said to be continuous at the number a if and only if the following three conditions are satisfied:
(i) Values of function exist, i.e. f\left( a \right) exists.
(ii) Limit of given function exist and finite. i.e. \mathop  {\lim }\limits_{x \to a} f\left( x \right) exist and finite.
(iii) Now check values of function and limit of function must be equal. i.e.

\mathop  {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)

If one or more of these three conditions fails to hold at a, the function f is said to be discontinuous at a.