Continuity at a Point via Pictures - At A Glance

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 x = c 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 x = c, 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 = c 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 .

Examples

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

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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.