The document discusses continuity of functions at a point c. It defines continuity as a function being defined at c, the limit existing at c, and the function value at c equaling the limit value. Five examples are given and categorized as continuous, discontinuous with removable discontinuities, or discontinuous with nonremovable discontinuities. Removable discontinuities can be resolved by redefining the function value at c, while nonremovable discontinuities cannot be resolved.

1. Continuity

2. So the 5 picturesof these are
In which below, let’s look at:
a. Is the function defined at c ?
continuous?
b. Does the limit exist at c ?
1 2
a. f(c) is und
c a. f(c) is und
c
b. lim DNE b. lim EXISTS
3 4 5
a. f(c) cis defined a. f(c) c defined
is a. f(c) is defined
c
b. lim DNE b. lim EXISTS b. lim EXISTS

3. 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.
* All three of these must occur.

4. 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.

5. Based on the definition of continuity,
which of the functions are
continuous?
a b
discontinuous discontinuous
c d e
discontinuous discontinuous continuous

6. 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.

7. In the 5 pictures below, let’s now
identify which have removable and
nonremovable discontinuities.
a b
nonremovable removable
c d e
nonremovable removable continuous