Book: RD Sharma - Mathematics (Volume 1)
Chapter: 9. Continuity
Subject: Maths - Class 12th
Q. No. 6 of Exercise 9.1
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If 
Find whether f is continuous at x = 0.
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.
