Unformatted text preview: Preliminary Exam  January 2002
__ 1._Let A and B be subsets of a metric sp_ace._Prove that A n B C
A08 and give anexample when AﬂByéAﬂB. 2. Let f and f’ be continuous functions on R. Prove that that the
sequence of functions f(9=+ 1/71) f($)
l/n converges to f’(:1:) uniformly on every interval [a, b], 00 < a < b < oo.
3. Let f be a Riemann integrable function on [0,1] and 971(3) = 17(3) = / f(t) dt. 3.) Show that there is a constant C such that F(z) —F(y) g CIz—y]
for every 2:, y 6 [0,1].
b) Give an example of f such that F is not diﬁerentiable at some
point.
4. Show that the sequence
tan—1012:) fn(l‘) = ﬂ is equicontinuous on R and converges uniformly to f (2:) = limnnoo fn(a:).
Show that f,’1(x) does not converge uniformly to f’ (x) 5. Determine the values of a for which f is differentiable at (0,0)
when 2 2 a ' 1 . f($,y) = (x + y ) SID. my: (way) :5 (0’0): 0) (33,3!) — (0,0).
6. Show that if (ﬂy) is a continuously differentiable function on
(—a, a), a > 0, such that ¢(0) = 0 and q5’(y) S k < 1 on (—a, a), then there is a > 0 and a unique differentiable function g on (—5,5)
satisfying the equation x = 9(1‘) + ¢(g(r)). ...
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