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.

Let’s solve :

To check whether function is continuous at x=0 we need to check whether LHL = RHL = f(c)

As continuity is to be checked at x = 0 therefore c=0.

As, …… say it equation 2

Clearly, f(0) = 1 using eqn 3

LHL =

Using equation 3 –

[∵ h is very small , is very large ‘–ve‘]

RHL =

Using equation 3 –

[∵ h is very small , is very large ‘+ve‘]

Clearly, LHL ≠ RHL ≠ f(0)

∴ We can easily say that f(x) is discontinuous at origin.