# Continuity

A function f (x) is said to be continuous at x = a, if and only if:

Note:

• The sum, difference and product of two continuous functions is a continuous function.

This property holds good for any finite number of functions.
• The quotient of two continuous functions is a continuous function provided the denominator is not equal to zero.

# Discontinuous function

A function f (x) is said to be discontinuous at x = a for any of the following reasons:
•  exists but it is not equal to f(a)
•  does not exist
• f is not defined at x = ai.e.f(a) does not exist
• If a function f (x) is not continuous at x = a, it is known as a discontinuous function.

Example
The value of f(0) so that f(x= (1 5x)1/x is continuous at x = 0 is
Solution

Example
Solution

But f(2) = k {f(x) = k when x = 2}
As f(x) is continuous at x = 2,

Example
Examine the function  for continuity at x = 2.
Solution
Here

So the right hand and left hand limits of f(x) as x â†’ 2 are the same.

But the value of f(x) at x = 2 is 6 i.e.f(2) = 6

âˆ´ The function is not continuous at x = 2