Left hand limit of f at 0 is
0
0
lim
( )
lim (– )
0
x
x
f x
x
−
−
→
=→
=
Similarly, the right hand limit of f at 0 is
0
0
lim
( )
lim
0
x
x
f x
x
+
+
→
=→
=
Thus, the left hand limit, right hand limit and the value of the function coincide at
x = 0 Hence, f is continuous at x = 0 Example 4 Show that the function f given by
f(x) =
3
3, if
0
1,
if
0
x
x
x

+
≠

=

is not continuous at x = 0 Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
107
Solution The function is defined at x = 0 and its value at x = 0 is 1