ANALYSIS I: PROBLEM SET # 8
DUE FRIDAY, 29 APRIL
Definition. Suppose X a topological space, and suppose Y a metric space; we’ll say that a sequence ( fn )n≥0 of funcY converges nonuniformly if the sequence ( fn )n≥0 does not converge uniformly, but there nevertheless
tions X
Y such that for every x ∈ X , one has fn (x) → f (x).
exists a function f : X
Construct explicit examples of each of the following, or else prove that no such example exists.
Exercise 62. A Lipschitz differentiable function f on [0, 1] such that f 0 attains neither a maximum nor a minimum
on [0, 1].
Exercise 63. A monotonic smooth function f such that lim x→∞ f (x) = 0, but lim x→∞ f 0 (x) 6= 0.
Exercise 64. A uniformly convergent sequence of compactly supported, continuous functions R
is not compactly supported.
Exercise 65. A continuous function f : R
uniformly to 0, but lim x→∞ f (x) 6= 0.
R whose limit
R such that if fn (x) = f (x + n), then the sequence ( fn )n≥0 converges
Exercise 66. A sequence ( fn )n≥0 of functions R
no fn is continuous at any point of R.
R that converge uniformly to a smooth function, even though
Exercise 67. A sequence of real-valued smooth functions on a compact set K ⊂ R that converges nonuniformly to
a continuous function.
Exercise 68. A sequence of real valued smooth functions on a open set U ⊂ R that converges nonuniformly on U
and converges uniformly on every closed interval I ⊂ U .
Exercise 69. Given a Gδ set E ⊂ R, a sequence of continuous functions R
set of points of continuity is exactly E.
Exercise 70. A sequence of functions R
uniformly.
R that converge to a function whose
R that converges nonuniformly with a subsequence that converges
Exercise 71. A nonzero integrable function on [0, 1] such that for any n ≥ 0,
Z1
f (x)x n d x = 0.
0
Exercise 72. A nonzero continuous function on [0, 1] such that for any n ≥ 0,
Z1
f (x)x n d x = 0.
0
1