Let c be any real number Then
lim
( )
x
c f x
→
 = lim
x
c k
k
→
=
Since f(c) = k = lim
x
c
→  f(x) for any real number c, the function f is continuous at
every real number Example 6 Prove that the identity function on real numbers given by f(x) = x is
continuous at every real number Solution The function is clearly defined at every point and f (c) = c for every real
number c