- Give an example of a continuous function defined on which does not achieve
its maximum on
- Give an example of a continuous function defined on which is bounded but
which does not achieve either its maximum or its minimum.
- Give an example of a discontinuous function defined on which is bounded
but does not achieve either its maximum or its minimum.
- Give an example of a continuous function defined on [0,1)∪(1,2] which is positive
at 2, negative at 0 but is not equal to zero for any value of x.
- Let f =
x5 +ax4 +bx3 +cx2 +dx+e where a,b,c,d, and e are numbers. Show
there exists real x such that f = 0
- Give an example of a function which is one to one but neither strictly increasing
nor strictly decreasing.
- Show that the function f =
xn − a, where n is a positive integer and a is a
number, is continuous.
- Use the intermediate value theorem on the function f =
x7 − 8 to show
must exist. State and prove a general theorem about
nth roots of positive numbers.
- Prove is irrational.
Hint: Suppose =
p∕q where p,q are positive integers
and the fraction is in lowest terms. Then 2q2 = p2 and so p2 is even. Explain why
p = 2r so p must be even. Next argue q must be even.
- Let f =
x − for
x ∈ ℚ, the rational numbers. Show that even though
< 0 and f
> 0, there is no point in ℚ where f = 0
. Does this contradict
the intermediate value theorem? Explain.
- It has been known since the time of Pythagoras that is irrational. If you throw
out all the irrational numbers, show that the conclusion of the intermediate value
theorem could no longer be obtained. That is, show there exists a function which
starts off less than zero and ends up larger than zero and yet there is no number
where the function equals zero.
Hint: Try f =
x2 −2. You supply the details.
- A circular hula hoop lies partly in the shade and partly in the hot sun. Show
there exist two points on the hula hoop which are at opposite sides of the hoop
which have the same temperature. Hint: Imagine this is a circle and points are
located by specifying their angle, θ from a fixed diameter. Then letting T be the
temperature in the hoop,
. You need to have T =
θ. Assume T is a continuous function of θ.
- A car starts off on a long trip with a full tank of gas. The driver intends to drive
the car till it runs out of gas. Show that at some time the number of miles the car
has gone exactly equals the number of gallons of gas in the tank.
- Suppose f is a continuous function defined on which maps
Show there exists x ∈ such that
x = f
. Hint: Consider h
and the intermediate value theorem. This is a one dimensional version of the
Brouwer fixed point theorem.
- Let f be a continuous function on such that
. Let n be a
positive integer. Show there must exist c ∈ such that
Hint: Consider h
−f. Consider the subintervals
k = 1,
,n− 1. You want to show that h equals zero on one of these intervals. If
h changes sign between two successive intervals, then you are done. Assume then,
that this does not happen. Say h remains positive. Argue that f
> f =
. It follows that h
< 0 but
- Use Theorem 6.5.5 and the characterization of connected sets in ℝ to give a quick
proof of the intermediate value theorem.
- A set is said to be totally disconnected if each component consists of a single point.
Show that the Cantor set is totally disconnected but that every point is a limit
point of the set. Hint: Show it contains no intervals other than single points.
- A perfect set is a non empty closed set such that every point is a limit point. Show
that no perfect set in ℝ can be countable. Hint: You might want to use the fact
that the set of infinite sequences of 0 and 1 is uncountable. Show that there is a
one to one mapping from this set of sequences onto a subset of the perfect set.
- Suppose f : K → ℝ where K is a compact set and f is continuous. Show
that f achieves its maximum and minimum by using Theorem 6.3.4 and the
characterization of compact sets in ℝ given earlier which said that such a set is
closed and bounded. Hint: You need to show that a closed and bounded set in
ℝ has a largest value and a smallest value.