4
Fig  5 5
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
111
Example 12 Discuss the continuity of the function defined by
f(x) =
2, if
0
2, if
0
x
x
x
x
+
<

− +
>

Solution Observe that the function is defined at all real numbers except at 0 Domain
of definition of this function is
D1 ∪ D2 where  D1 = {x ∈ R : x < 0} and
      D2 = {x ∈ R : x > 0}
Case 1 If c ∈ D1, then lim
( )
lim
x
c
x
c
f x
→
=→
 (x + 2)
= c + 2 = f (c) and hence f is continuous in D1 Case 2 If c ∈ D2, then lim
( )
lim
x
c
x
c
f x
→
=→
 (– x + 2)
= – c + 2 = f (c) and hence f is continuous in D2