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Continuity of Functions

Continuity of Functions

At a Glance - Continuity at a Point via Formulas

It's good to have a feel for what continuity at a point looks like in pictures. However, sometimes we're asked about the continuity of a function for which we're given a formula, instead of a picture. When this happens, remember that the following three statements must all hold for f to be continuous at c.

  1. I. The function f is defined at x = c.
      
  2. The limit  exists.
      
  3. The value f(c) agrees with the limit 
      

Example 1

Determine whether the function 

is continuous at x = 1.


Example 2

Determine whether the function 

 

 is continuous at x = 2.


Example 3

Determine whether the function 

 

is continuous at x = 0.


Example 4

Determine whether the function 

is continuous at x = 0.


Example 5

At what values is f discontinuous? 


Exercise 1

Determine whether the function

  is continuous at each given value. If not, explain.

  • x = -5  
  • x = -4  
  • x = 0  
  • x = 2  
  • x = 3  

Exercise 2

For what values of x is the function discontinuous.


Exercise 3

For the function, determine all values at which the function is discontinuous.


Exercise 4

For what values of x is h(x) discontinuous.


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