##### Discuss the continuity of the function at the point x = 1/2.

Ideas required to solve the problem:

1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x) , we can say that if we plot the coordinates (x , f(x)) and try to join all those points in the specified region, we can do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve without any breakage.

Mathematically we define the same thing as given below:

A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to be checked

If:– …… equation 1

where h is a very small positive no (can assume h = 0.00000000001 like this )

It means :–

Limiting value of the left neighbourhood of x = c also called left hand limit LHL must be equal to limiting value of right neighbourhood of x= c called right hand limit RHL and both must be equal to the value of f(x) at x=c i.e. f(c).

Thus, it is the necessary condition for a function to be continuous

So, whenever we check continuity we try to check above equality if it holds true, function is continuous else it is discontinuous.

Given, …… equation 2

we are asked to check its continuity at x=1/2

we need to check LHL ,RHL and value of function at x = 1/2 ,if all comes out to be equal we can say f(x) is continuous at x=1/2 else it is discontinuous.

Clearly,

f( ) = [from eqn 2]

LHL = RHL = Thus, LHL = RHL = f(0)

f(x) is continuous at x = 16