Suppose f is a real-valued continuously differentiable function on [a,b] with f(a) = f(b) = 0 and
∫ab f2(x) dx = 1.
Show that ∫ab x f(x) f'(x) dx = -1/2.
Prove that if f ∈ ℜ on [a,b] and g is a function for which g(x) = f(x) for all but
finitely many x, then g ∈ ℜ on [a,b] (Note: in this case,
∫ab g dx = ∫ab f dx).
Does this result still hold if g(x) = f(x)
for all but countably many x?
Let f : [0, ∞) → R be defined as f(x) = 0 if 0 ≤ x ≤ 1/2 and f(x) = 1
if 1/2 < x < ∞. Show that the function F(x) = ∫0x f(t) dt, defined for
0 ≤ x < ∞, is differentiable for x ≠ 1/2 and is not differentiable at x = 1/2.
Exam 1 on Friday, January 30 covers Chapters 5 and 6.
The converse of Theorem 7.10 is false. Exhibit an example to illustrate this.
Be sure to clearly explain how your example shows the converse to be false.
Prove that the family {sin(nx) | n ∈ N} is not equicontinuous on the interval [-1,1].
Prove that the family of all polynomials of degree less than or equal to N with coefficients in [-1,1] is
uniformly bounded and equicontinuous on any compact (closed and bounded) interval.
Give an example of a metric space X and a sequence of functions {fn} on X such that
{fn} is equicontinuous but not uniformly bounded.
Give an example of a uniformly bounded and equicontinuous sequence of functions on R which does not
have any uniformly convergent subsequences.
In 19, you may use (without proof) the fact that in any metric space, sequential compactness is equivalent to compactness. A metric space S is sequentially compact if and only if every sequence in S contains a subsequence that converges to a point in S.
You may use the results of exercises 6.2 and 6.12.
Exam 2 on Friday, February 27 covers Chapters 5 - 7, with an emphasis on Chapter 7.
(Corrected) For any continuous function f on [0,1], let F[f](x) = ∫0x f(t) dt. Show that the set of functions S = {F[f] | ||f|| ≤ 1} is uniformly bounded and equicontinuous.
Suppose f : [0,π] → R is continuous and ∫0π f(x) sin(nx) dx = 0 for every integer n ≥ 1. Does it follow that f = 0? Prove or exhibit a counterexample.
If f is real analytic in a neighborhood of x0 and f(x0) = 0, show that f(x)/(x - x0) is real analytic in the same neighborhood.
Prove that if f(x) is real analytic on (a,b) and c ∈ (a,b), then F(x) = ∫cx f(t) dt is also real analytic on (a,b).
Prove that xn → 0 with respect to the L2 metric on [0,1], but not in the uniform metric. {Note: the uniform metric is the metric induced by the supremum norm on C([0,1])}.