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Continuity of Functions

Continuity of Functions

At a Glance - Continuity at a Point via Pictures

The Pencil Rule of Continuity

A continuous function is one that we can draw without lifting our pencil, pen, or Crayola crayon.

Here are some examples of continuous functions:

If a function is continuous at xc we can start with our pencil a little to the left of x = c and trace the graph until our pencil is a little to the right of xc, without lifting our pencil along the way.

We will now return to those functions that are continuous at x = c. We can trace each function with a pencil, from one side of x = to the other, without lifting the pencil.

If a function isn't continuous at x = c, we say it's discontinuous at x = c.

Sample Problem

This function is not continuous at x = c, since the function isn't even defined at x = c. We can't compare the value of f(c) to , since neither exists!

Sample Problem

This function "jumps" at x = c. To draw the graph we would have to draw one line, stop at x = c and lift the pencil, then draw another line. As far as the limit definition goes,  doesn't even exist (the one-sided limits disagree). Therefore f can't possibly be continuous at c.

Sample Problem

This function also jumps at x = c. To draw the graph we would have to draw a line, lift the pencil and draw a dot at x = c, then lift the pencil again to draw the remaining line. In this graph both f(c) and exist, but the function value disagrees with the limit.

If a function f is discontinuous at x = c, then at least one of three things need to go wrong. Either

  • f(c) is undefined (therefore we can't draw it at all),
      
  • we need to move the pencil either just before or just after we reach x = c ( doesn't exist), or
      
  • we need to move the pencil either just before or just after we reach x = c and we need to draw a separate little dot for f(c).
      

In other words: for a function f(x) to be continuous at x = c, three things need to happen:

  • The function must be defined at x = c (that is, f(c) must exist)
      
  • The limit must exist (both one-sided limits must exist and agree)
      
  • The value f(c) must agree with the limit .
      

Example 1

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

Example 2

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
      

Example 3

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
      

Example 4

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
      

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