We
also know that
lim
( )
x
c f x
→
 =
3
2
3
2
lim (
1)
1
x
c x
x
c
c
→
+
−
=
+
−
Thus lim
( )
( )
x
c f x
f c
→
=
, and hence f is continuous at every real number This means
f is a continuous function Example 9 Discuss the continuity of the function f defined by f (x) = 1
x , x ≠ 0 Solution Fix any non zero real number c, we have
1
1
lim
( )
lim
x
c
x
c
f x
x
c
→
=→
=
Also, since for c ≠ 0, 
1
f c( )
=c
, we have lim
( )
( )
x
c f x
f c
→
=
 and hence, f is continuous
at every point in the domain of f