Book: RD Sharma - Mathematics (Volume 1)
Chapter: 9. Continuity
Subject: Maths - Class 12th
Q. No. 2 of Exercise 9.2
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Discuss the continuity of the function 
.
A real function f is said to be continuous at x = c, where c is any point in the domain of f if :
where h is a very small ‘+ve’ no.
i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.
This is very precise, using our fundamental idea of the limit from class 11 we can summarise it as
A function is continuous at x = c if :
Here we have,
…….equation 1
The function is defined for all real numbers, so we need to comment about its continuity for all numbers in its domain ( domain = set of numbers for which f is defined )
Function is changing its nature (or expression) at x = 0, So we need to check its continuity at x = 0 first.
NOTE: 
LHL = 
= 
= 
[using eqn 1 and idea of mod fn]
RHL = 
[using eqn 1 and idea of mod fn]
f(0) = 0
[using eqn 1]
Clearly, LHL ≠ RHL ≠ f(0)
∴ function is discontinuous at x = 0
Let c be any real number such that c > 0
∴ f(c) = 
[using eqn 1]
And, 
Thus, 
∴ f(x) is continuous everywhere for x > 0.
Let c be any real number such that c < 0
∴ f(c) = 
[using eqn 1 and idea of mod fn]
And, 
Thus, 
∴ f(x) is continuous everywhere for x < 0.
Hence, We can conclude by stating that f(x) is continuous for all Real numbers except zero.
