5.3 Continuity on Compact Sets and Intervals 255
- Prove by example that
(a) the continuous image of a bounded set need not be bounded.
(b) the continuous image of a closed set need not be closed.
- Suppose f : A--+JR is continuous on a closed set A. Prove that '</ c E JR,
the set {x EA: f(x) = c} is closed.
- Suppose f, g : A--+JR are continuous on a closed set A. Prove that the set
{x EA: f(x) = g(x)} is closed.
- Suppose f is positive and continuous on some compact set A. Prove that
f is bounded away from 015 on A.
- Prove the intermediate value theorem (Corollary 5.3.9).
11. Suppose f is continuous on [a,b] and 't:/x E [a,b], 3y E [a,b] 3 lf(y)I:::;
~lf(x)j. Prove that 3xo E [a,b] 3 f(xo) = 0. [Hint: use sequences.]
- Suppose f : I ---+ JR is continuous on an interval I and j(a)f(b) < 0 for
some a, b E J. Prove that f(x) = 0 for some x between a and b.
- Use the location of roots principle to find a root of the equation 4x^3 -
5x^2 + x - 7 = 0, correct to three decimal places.
- Use the lo cation of roots principle to find two roots of the equation x^4 -
x^3 - 10 = 0, correct to three decimal places.
15. Prove that a polynomial function of odd degree has at least one real root.
[Hint: Use Theorem 4.4.24 and the intermediate value theorem.]
- Prove that a polynomial function p(x) = a 0 + a 1 x + · · · + anxn of even
degree n, in which aoan < 0, must have at least two real roots, one
positive and one negative. [See hint for Exercise 16.J
- Prove that cos x = x for some x E ( 0, ~).
- Prove that x2x = 1 for some x E (0, 1). [Assume 2x is co ntinuous every-
where.]
- Suppose f ,g: [a,b]---+ JR are continuous, f(a) < g(a) and f(b) > g(b).
Prove that f(c) = g(c) for at least one c E (a, b).
- See Definition 4.2.8.
