ContinuityContinuity

21

3 4 5

In the 5 pictures below, let’s look at:a. Is the function defined at c ?b. Does the limit exist at c ?

cc

c c c

21

3 4 5

In the 5 pictures below, let’s look at:a. Is the function defined at c ?b. Does the limit exist at c ?

a. f(c) is undb. lim DNE

a. f(c) is undb. lim EXISTS

a. f(c) is definedb. lim DNE

a. f(c) is definedb. lim EXISTS

a. f(c) is definedb. lim EXISTS

cc

c c c

21

3 4 5

In the 5 pictures below, let’s look at:a. Is the function defined at c ?b. Does the limit exist at c ?

a. f(c) is undb. lim DNE

a. f(c) is undb. lim EXISTS

a. f(c) is definedb. lim DNE

a. f(c) is definedb. lim EXISTS

a. f(c) is definedb. lim EXISTS

cc

c c c

So which of these are continuous?

* All three of these must occur.

A function is continuous at c if…

• 1. The function is defined at c.• 2. The limit exists at c.• 3.The value of function at c equals

the value of limit at c.

Another way of saying this is that a function is

continuous at every point in the interval if you can

draw it without lifting your pencil.

ba

c d e

Based on the definition of continuity, which of the functions

are continuous?

ba

c d e

Based on the definition of continuity, which of the functions

are continuous?

discontinuous discontinuous

discontinuous discontinuous continuous

Removable vs. Nonremovable Discontinuities

• A discontinuity at c is called removable if f can be made continuous by appropriately defining (or redefining) f(c).

• If you can simply “plug up the hole”, the discontinuity is removable.

• A discontinuity is nonremovable if there is no way to define the function at a point to make it continuous.

ba

c d e

In the 5 pictures below, let’s now identify which have removable and

nonremovable discontinuities.

ba

c d e

In the 5 pictures below, let’s now identify which have removable and

nonremovable discontinuities.

nonremovable removable

nonremovable removable continuous