Additional problems for midterm
February 3, 2009
1. If fand gare both uniformly continuous functions from Rto R,
(a) Is the product function f๎
guniformly continuous?
(b) What happens if both fand gare bounded?
2. Show that: f:RโRis not uniformly continuous if and only if there is
an ๎ > 0 and there are sequences (xn) and (yn) such that |xnโyn|<1/n
and |f(xn)โf(yn)| โฅ ๎.
3. A function f:SโRโRis Lipschitz in Sif there is a constant
Lโฅ0 such that |f(x)โf(y)| โค L|xโy|for every x,yin S.
(a) Show that a Lipschitz function in Sis uniformly continuous on S.
(b) Find a bounded continuous function f:RโRthat is not uni-
formly continuous and therefore not Lipschitz.
4. Show that if f:RโRis continuous and BโRis bounded, then
f(B) is bounded.
5. Prove that f(x) = p|x|is continuous. (Use any theorem).
6. Verify Theorem 18.1 for
(a) f(x) = x/(x2+ 1) on [0,1].
(b) f(x) = x3โxon [โ1,1].
7. Let fn:RโRbe given by fn(x) = sin(x)/n. Prove that fnโ0
uniformly.
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