106 CHAPTER 4. CONTINUITY AND LIMITS

4.6 Exercises1. Give an example of a continuous function defined on (0,1) which does not achieve

its maximum on (0,1) .

2. Give an example of a continuous function defined on (0,1) which is bounded butwhich does not achieve either its maximum or its minimum.

3. Give an example of a discontinuous function defined on [0,1] which is bounded butdoes not achieve either its maximum or its minimum.

4. Give an example of a continuous function defined on [0,1)∪ (1,2] which is positiveat 2, negative at 0 but is not equal to zero for any value of x.

5. Let f (x) = x5 + ax4 + bx3 + cx2 + dx+ e where a,b,c,d, and e are numbers. Showthere exists real x such that f (x) = 0.

6. Give an example of a function which is one to one but neither strictly increasing norstrictly decreasing.

7. Show that the function f (x) = xn−a, where n is a positive integer and a is a number,is continuous.

8. Use the intermediate value theorem on the function f (x) = x7 −8 to show 7√

8 mustexist. State and prove a general theorem about nth roots of positive numbers.

9. Prove√

2 is irrational. Hint: Suppose√

2 = p/q where p,q are positive integers andthe fraction is in lowest terms. Then 2q2 = p2 and so p2 is even. Explain why p = 2rso p must be even. Next argue q must be even.

10. Let f (x) = x−√

2 for x ∈ Q, the rational numbers. Show that even though f (0) <0 and f (2) > 0, there is no point in Q where f (x) = 0. Does this contradict theintermediate value theorem? Explain.

11. A circular hula hoop lies partly in the shade and partly in the hot sun. Show thereexist two points on the hula hoop which are at opposite sides of the hoop whichhave the same temperature. Hint: Imagine this is a circle and points are located byspecifying their angle, θ from a fixed diameter. Then letting T (θ) be the temperaturein the hoop, T (θ +2π) = T (θ) . You need to have T (θ) = T (θ +π) for some θ .Assume T is a continuous function of θ .

12. A car starts off on a long trip with a full tank of gas which is driven till it runs out ofgas. Show that at some time the number of miles the car has gone exactly equals thenumber of gallons of gas in the tank.

13. Suppose f is a continuous function defined on [0,1] which maps [0,1] into [0,1] .Show there exists x ∈ [0,1] such that x = f (x) . Hint: Consider h(x)≡ x− f (x) andthe intermediate value theorem. This is a one dimensional version of the Brouwerfixed point theorem.

14. Let f be a continuous function on [0,1] such that f (0) = f (1) . Let n be a positiveinteger larger than 2. Show there must exist c ∈

[0,1− 1

n

]such that f

(c+ 1

n

)=

f (c). Hint: Consider h(x) ≡ f(x+ 1

n

)− f (x). Consider the subintervals

[ k−1n , k

n

]