Book: RD Sharma - Mathematics (Volume 1)
Chapter: 9. Continuity
Subject: Maths - Class 12th
Q. No. 10 of Exercise 9.2
Listen NCERT Audio Books to boost your productivity and retention power by 2X.
Discuss the continuity of f(x) = sin |x|.
Basic Idea:
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 :
NOTE: 
Here we are given with
f(x) = sin |x|
…………………………equation 1
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.
LHL = 
= 
[using eqn 1]
RHL = 
[using eqn 1]
f(0) = 
[using eqn 1]
Clearly, LHL = RHL = f(0)
∴ function is continuous at x = 0
For all x ≠ 0, f(x) is simply a trigonometric function which is everywhere continuous which can be verified by seeing its plot in X–Y plane or even can be verified using limit.
∴ f(x) is everywhere continuous in its domain i.e. it is continuous for all real values of x
