Preliminary Analysis Exam
August 24, 2009
1. If F1 and F2 are closed subsets of R1 and dist(F1 , F2 ) = 0 then F1 F2 = . Prove or
give a counterexample.
2. Newtons method for nding zeroes of a function f : R1 R1 is based on the recursion
formula
f (xn )
xn+1 = xn
, n 1.
f (xn )
Show that if f C 1 , f (a) = 0 and f (a) = 0, then there exists a > 0 such that if
|x1 a| < then xn a. (Suggestion: use the Mean Value Theorem.)
3. Let f : [0, ) [0, ) and for h > 0 and k 1 set
Mk (h) =
sup
f (x), mk (h) =
(k1)hx<kh
Let
inf
(k1)hx<kh
f (x).
U ( h) =
Mk (h)h, L(h) =
k=1
mk (h)h.
k=1
We say f is directly Riemann integrable if U (h) < for all h > 0 and
lim(U (h) L(h)) = 0.
h0
Recall f is improperly Riemann integrable on [0, ) if f is Riemann integrable on
[0, a] for every a > 0, and
a
lim
a
f (t) dt < Show .
0
(a) that if f is continuous and nonincreasing, then f is directly Riemann integrable
whenever f is improperly Riemann integrable on [0, ).
(b) Give an example of a continuous function f which is improperly Riemann integrable
on [0, ) but not directly Riemann integrable.
4. Suppose f : [0, ) [0, ) is such that for any sequence an of nonnegative terms we
have
an < =
n=1
f (an ) <
n=1
Prove that
lim sup
x0+
f (x)
<
x
5. Let f be a continuous real valued function dened on the unit square and for each
0 x 1 let fx be the function on the unit interval dened by fx (y ) = f (x, y ). Prove
that for any sequence xn in [0,1] there is a subsequence nk such that fxnk converges
uniformly on [0,1].
6. If c is a real parameter prove that x7 + x + c = 0 has a unique real root and that this
root is a dierentiable function of c.