Continuity at a Point via Pictures Examples

Look at the graph of the function f(x).

The following three statements must all hold for f to be continuous at c.

• The function f is defined at x = c.
    
• The limit  exists.
    
• The value f(c) agrees with the limit .

For each given value of c, determine whether each of the three statements holds. Use this to determine whether f is continuous at the given value of c.

• c = -2
    
• c = -1
    
• c = 0
    
• c = 2
    
• c = 5

• c = -2

Statement I holds (f(-2) exists).

Statement II does not hold since the one-sided limits of f as x approaches -2 do not agree the left limit is f(-2), while the right limit is -∞.

Statement III does not hold since  does not exist. 

To sum it up, f is not continuous at x = -2.

• c = -1

Statements I, II, and III all hold. The function f is continuous at x = -1.

• c = 0

Statement I does not hold (f(0) does not exist). Statement II holds. Since f(0) does not exist,

Statement III does not hold. The function f is not continuous at x = 0.

• c = 2

Everything goes wrong here. Statements I, II, and III all fail. The function f is not continuous at x = 2.

• c = 5

Statements I, II, and III all hold. The function f is continuous at x = 5.

Look at the graph of the function f(x).

The following three statements must all hold for f to be continuous at c.

• The function f is defined at x = c.
    
• The limit  exists.
    
• The value f(c) agrees with the limit .

For each given value of c, determine whether each of the three statements hold. Use this to determine whether f is continuous at the given value of c.

• c = -2
    
• c = -1
    
• c = 0
    
• c = 1
    
• c = 2
    

• c = -2
  All statements hold. The function f is continuous at x = -2.

• c = -1
  Statement I holds. Statement II does not hold because the one-sided limits of f disagree at x = -1. Statement III does not hold because not exist. The function f is not continuous at x = -1.

• c = 0
  All statements hold. The function f is continuous at x = 0.

• c = 1
  Statement I holds. Statement II does not hold because the one-sided limits of f disagree at x = 1. Statement III does not hold because  does not exist. The function f is not continuous at x = 1.

• c = 2
  All statements fail. The function f is not continuous at x = 2.

Notice that the third condition for continuity relies on the other two. If the statement "The function f is defined at x = c" fails, or if the statement "The limit  exists" fails, then we can't possibly have

Look at the graph of the function f(x).

Determine whether the function f is continuous at each given value. If not, explain.

• x = -2
    
• x = 0
    
• x = 2
    
• x = 4
    
• x = 5
    

• x = -2
    
  The function f is not continuous at x = -2 because f(-2) is undefined. There is a hole in the graph when x = -2.

• x = 0
    
  The function f is not continuous at x = 0 because  does not exist. The one-sided limits at x = 0 disagree.

• x = 2
    
  The function f is continuous at x = 2 because f(2) is defined and agrees with . We can trace the graph with a pencil from one side of x = 2 to the other side.

• x = 4
    
  The function f is not continuous at x = 4 because f(4) is undefined. The limit  doesn't exist either.

• x = 5
    
  The function f is continuous at x = 5 because f(5) is defined and agrees with . We can trace the graph with a pencil from one side of x = 5 to the other side.

Look at the graph of the function g(x).

Determine whether the function g is continuous at each given value. If not, explain.

• x = -20
    
• x = -10
    
• x = 0
    
• x = 5
    
• x = 10
    

• x = -20
    
  Yes. The function g is continuous at x = -20 because we can trace that part of the graph without lifting the pencil. More formally, g(20) exists and agrees with . The function doesn't do anything strange at x = -20.

• x = -10
    
  No. The function g is not continuous at x = -10 because g(-10) does not exist.  doesn't exist either.

• x = 0
    
  Yes. The function g is continuous at x = 0 because we can trace that part of the graph without lifting the pencil. More formally, g(0) exists and is equal to .

• x = 5
    
  No. The function g is not continuous at x = 5 because g(5) and   disagree.

• x = 10
    
  No. The function g is not continuous at x = 10 because   does not exist (the one-sided limits disagree).