On-line Math 21

1.1  Continuity Examples

1. Let
   

   f(x) = ì í î 3x+1, if x ³ 1 x+2, if x < 1 .

   Then f is not continuous at a = 1 . This is because
   

   lim x® 1  f(x)

   does not exist, which you should recall from an earlier lecture.
2. Let
   

   f(x) = ì í î x2, if x ¹ 1 5, if x = 1 .

   This function is also not continuous at x = 1 , since
   

   lim x® 1  f(x) = 1,

   even though f(1) = 5 . The difference here is that the function could have been continuous, if it weren't for the fact that it was ``poorly'' defined at x = 1 .
3. Let
   

   f(x) = ì í î 3x+1, if x ³ 1 x+3, if x < 1 .

   Then f is continuous at a = 1 . This is because
   

   lim x® 1  f(x)

   does exist, and is the same as f(1) .
4. Since, for any polynomial f(x) = a_nxⁿ+... +a₁x+a₀ , and for any x = a ,
   

   lim x® a

   f(x) = f(a)

   
   , any polynomial is continuous. We say that a function f is continuous, instead of continuous at a point or on an interval, if it is continuous at every point of its domain.
5. f(x) = |x| is continuous, even though it has ``bad behavior'' at x = 0 . Its behavior is just not that bad there. The limit of f(x) , as x approaches 0, is 0, and since that is the value of the function, then it is continuous there. At other points, f is just like a polynomial in some open interval (either x or -x ), so it is continuous.

Copyright (c) by David L. Johnson, last modified
On 18 Apr 2000, 10:19..