# Continuity of Functions

# 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 *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 *f*(*c*) to doesn't even exist!

### 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 we can't possibly have

### 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 .