# 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.
- 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*=*a*,*i.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 + 5*x*)^{1/x}is continuous at*x*= 0 isSolution

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

But the value of

âˆ´ The function is not continuous at

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) = 6âˆ´ The function is not continuous at

*x*= 2