One sided Limits,
Continuity of Closed Intervals,
Continuity Theorems
, and IVT
Ex: If
and
then
Is the function
contin
uous over the interval [0,1]
Is the function
continuous over the interval [0,0.5]
Is the function f(x) =
continuous over the interval [2, 12]
f(x) = x
2
+ 2x
g(x) =
h(x) =
j(x) = sin(x)
Is f(x) + j(x) continuous?
Is
continuous?
Is f(g(x)) continuous at x =
2
?
Is f(h(x)) continuous at x = 0
?
Is j(h(x)) continuous at x =
?
Intermediate Value Theorem
Another way to look at the Intermediate Value Theorem: Let suppose the length of your
hair is currently 2 inches long. In one year your hair ends up being 6 inches long. The
IVT would state that there
must be a time were your hair was exactly 3 inches long.
We can use the IVT to approximate zeros of an equation. Lets say the function f(x) is
continuous and has the following x/y chart.
X:

3

2

1 0 1 2 3 4
Y: 2 3 1

1 2
3 4

2
Approximate where the x

intercepts for this continuous function would fall?
Textbook Example: Show that the IVT applies to the indicated interval and find the
value of c guaranteed by the theorem.
Ex.
,
,
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