Math 311
Dr. Sarah Raynor
May 5, 2016
SOME QUESTIONS THAT WILL BE ON THE
FINAL
1. Recall from calculus that the integral of a continuous function f : [a, b] → R is given
Rb
P
) b−a
.
by a f (x)dx = limn→∞ ni=1 f (a + (b−a)i
n
n
(a) (8 points) Prove that if (fn ) are a sequence of continuous functions on [a, b] that
Rb
Rb
converge to a function f on [a, b], then limn→∞ a fn (x)dx = a f (x)dx.
(b) (6 points) Give an example of a sequence of continuous functions (fn ) on [a, b]
Rb
that converge pointwise to a function f on [a, b] so that limn→∞ a fn (x)dx 6=
Rb
f (x)dx.
a
2. (3 points each) Provide an example of each or explain why it doesn’t exist.
(a) A function f that is continuous at every irrational number but discontinuous at
every rational number.
(b) A set that is totally disconnected and dense in R.
(c) A bijective map from the set of sequences a : N → {0, 1} to the set of real
numbers.
P
(d) A series of functions ∞
n=1 an (x) where each an is continuously differentiable that
converges uniformly on R but the sum function is nowhere differentiable.
P∞ P∞
P P∞
a
=
3
but
(e) A doubly indexed sequence (aij ) so that ∞
ij
j=1
i=1 aij =
i=1
j=1
−1.
3. (7 points) Consider two Dedekind cuts A and B so that A < 0 and B ≥ 0. Give a
reasonable definition for the Dedekind cut AB and prove that your definition yields a
valid cut.
4. ConsiderPan infinitely differentiable function f (x) on R. Consider also a power series
n
P (x) = ∞
n=0 an x .
(a) (5 points) Suppose that there is an R > 0 so that P (x) = f (x) for all x ∈ [−R, R].
Derive a formula for an in terms of f and its derivatives in this situation.
(b) (9 points) Prove that, under the conditions of part (a), f 0 (x) can also be represented by a power series on (−R, R) and find a formula for that series.
(c) (5 points) Is it necessarily true that if f (x) is infinitely differentiable and an
satisfies the formula from (a) then f (x) = P (x) on [−R, R] for some R > 0?
Either prove that it is or provide a counterexample.
5. (5 points) What was your favorite (mathematical) thing that you learned this semester
and why? Explain in full sentences.