Please hand in the following exercises from the text "An Introduction to Analysis, Third
Edition" by William Wade, Prentice Hall, 2004.
78 [ 2c, 4, 5 ]
Please hand in the following additional exercises.
Let I=[a,b] be a closed, bounded interval.
Suppose f : I --> R is continuous on I.
Suppose there is a real number M such that f(x) < M
for all x in I.
Prove that there is a real number R < M such that f(x) ≤ R
for all x in I.
Suppose f : I --> R is continuous on I.
Suppose that for every x in I there is y in I such that
2|f(y)| ≤ |f(x)|.
Prove that there is c in I such that f(c) = 0.
Suppose f : I --> I is continuous on I.
Prove that there is a p in I such that f(p) = p.
(Such a number p is called a fixed point of f.)