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Unformatted text preview: = 1/x is not uniformly continuous on (0, 1).
(You can use that a sequence is a Cauchy sequence if and only if it is convergent.)
9. (10: 6, 4) Suppose f (x) is a function on [a, b]; assume that f (a) < f (b). If f (x) is
continuous and strictly increasing, the following statement will be true:
For each point y in the interval [f (a), f (b)], there is one and only one point x in the
interval [a, b] such that y = f (x).
a) Prove this statement.
b) For each of the two hypotheses on f (x), tell what part of the statement will not be
true in general if that hypothesis is dropped, and give an example to illustrate this.
10. (10: 5, 5) a) Suppose f (x) is continuous on [a, b] and there is a sequence of distinct
points xn in [a, b] such that f (xn ) = 0 for all n.
Prove there is a subsequence of xn which converges to a point c in [a, b] such that f (c) = 0.
b) Continuing part (a), suppose that f ′ (x) exists on [a, b] and is continuous. Prove that
also f ′ (c) = 0.
11. (10) Suppose that f (x) is Riemann-integrable and f (x) = 1 whenever x is a rational
number. (Its value when x is an irrational number is not known.)
Evaluate, with proof, the integral
f (x) dx.
a 12. (5) For what values of k does ;the integral � ∞ 0+ 2 xk dx
x + x2...
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This note was uploaded on 01/22/2014 for the course MATH 18.100A taught by Professor Arthurmattuck during the Fall '12 term at MIT.
- Fall '12