Download Final Exam with Solution for Analysis I | MATH 521 and more Exams Advanced Calculus in PDF only on Docsity! 521 Analysis I Spring 2011 Final Exam
Keep justification short and to the point. The total is 100.
)= {2 €R:0< a2 <1} be the open unit interval in R. Give an example of
) such that lim 2, lies outside (0,1) but lim z, lies inside (0, 1).
1. (10 pts) Let (0,
a sequence {z,} in (0,
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2. (10 pts) Can there exist a continuous function f on [0,1] such that f is not constant and all
values f(x) are rational?
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wu be an vanakierad Im (u,v).
3. (15 pts) Let (X,d) be a metric space and fix a point w ¢ X. Use the triangle inequality to
show that the function f(x) = d(x, w) is uniformly continuous.
dlx,w) € d(my) + d(4,w) eo dlxyw)-d lg wF A (x,y)
dly,w) € dlyjx) +d(yw) =e dAlgw) — Abow) € dlmy),
“Toqethes thee awe Id (nw) Atyow)| Z d(x).
TT hes ' given E>0,) if A(my\ <€ then
[dGow) — d(yyw) | < €.
4, (15 pts) Let (X,d) be a metric space. Recall the definition of the distance of a point x to a
set A:
dist(z, A) = inf{d(z,a) : a € A}.
Suppose A is compact. Show that then there exists a point z € A such that d(x, z) = dist(z, A).
(z is a closest point to x in A.)
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7. (a) ( pts) State the definition of equicontinuity of a sequence of real-valued functions { fy} on
a metric space.
(b) (15 pts) Let {f,} be a sequence of real-valued functions on {0, 1] such that each f, is continuous
on {0, 1], differentiable on (0,1), fn(0) = 0, and (fi (x)| < 7 for all n and all x € (0,1). Prove that
there exists a subsequence of {f,} that converges uniformly on (0, 1].
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}$.69 ~ faq) 1 = | £(g)Q-y9)1 27 Ixy l,
. —_ &£ ‘IL werk.
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