Continuity of a Function

A moving physical object cannot vanish at one 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 a 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 the function exist, i.e. $$f\left( a \right)$$ exists.

(ii) The limit of the given function exists and is finite. i.e. $$\mathop {\lim }\limits_{x \to a} f\left( x \right)$$ exists and is finite.

(iii) The 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 $$a$$.