MATH 422/502 (072) TEST 2
1. Let (fn) be the sequence of functions defined by
fn(x) = nx
1 + n2x2, x ∈R.
(a) (10 pts) Show that fn→0 pointwise on R.
(b) (5 pts) Show that if x≥1, then fn(x)≤1/n.
(c) (10 pts) Use part (b) to give a careful ≤-Nproof that fn→0 uniformly on [1,1).
(d) (10 pts) Does fn→0 uniformly on [0,1]? Justify.
2. (15 pts) Let (fn) be a sequence of continuous functions defined on a set S⊂Rand let
f:S→Rbe a function such that fn→funiformly. Prove that fis continuous. (Recall that
Prof. Ross calls this the “famous ≤/3 argument”.)
3. (10 pts) Show that if P|ak|<1, then Pakxkconverges uniformly on [−1,1] to a continuous
function.
4. (10 pts) Let f(x) = x2for xrational and f(x) = 0 for xirrational. Prove that fis differentiable
at x= 0.
5. (10 pts) Prove that |cos x−cos y|≤|x−y|holds for all x, y ∈R.
6. (10 pts) Let f:R→Rbe a function that satisfies
|f(x)−f(y)|≤(x−y)2
for all x, y ∈R. Prove that fis a constant function. x
7. (10 pts) Let f: [0,1] →Rbe defined by f(1/2) = 1 and f(x) = 0 for x6= 1/2. Prove that f
is integrable and that R1
0f= 0.
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