# Application of the Uniform Continuity Theorem Questions

Question Description

I’m working on a mathematics multi-part question and need an explanation to help me study.

Sec 4.3
#2 (b) & (c)
#3 (c)
#4 (b) & (c)
#8
#10 ( Hint: Use proof by contradiction. Apply Bolzano-Weirstrass & problem 8 to get contradiction).

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EXERCISES 4.3
1. Prove Theorem 4.3.3.
2. Show that the following functions are not uniformly continuous on the given domain.
*a. f(x) = x. Dom f = (0.00) b. 8(x) =
c. h(x) = sin
3. Prove that each of the following functions is uniformly continuous on the indicated set.
*a. f(x) – Tx: +€ (0.00) b. 8(x) = x, XEN
– Domg = (0.00)
sin
Dom h = (0.00)
XER
d. k(x) = cos x, XER
c. h(x) = 2
6. e(x) =
sinx
€ (0.00) *. f(x) = € (0, 0)
4. Show that each of the following functions is a Lipschitz function.
•2. f(x) – Dom f = [0,00), a > 0 b. 8(x) = Dij Dom 8 = [0,00)
ch(x) = sin
sin Dom h = [0,00), a > 0 d. p(x) a polynomial, Domp = (-a, a), a > 0
5. a. Show that f(x) = Vx satisfies a Lipschitz condition on (a,00), a > 0.
b. Prove that is uniformly continuous on (0.00).
c. Show that does not satisfy a Lipschitz condition on (0,0).
6. Suppose ECR and f. 8 are Lipschitz functions on E.
a. Prove that f + g is a Lipschitz function on E.
b. If in addition f and g are bounded on E, or the set E is compact, prove that fg is a Lipschitz function on E.
7. Suppose ECR and f. 8 are uniformly continuous real-valued functions on E.
a. Prove that f + g is uniformly continuous on E.
*b. If, in addition, and g are bounded, prove that fg is uniformly continuous on E.
c. Is part (b) still true if only one of the two functions is bounded?
8. Suppose ECR and f: ER is uniformly continuous. If {xa} is a Cauchy sequence in E, prove that {f(x)} is a
Cauchy sequence
9. Let f:(a,b)-R be uniformly continuous on (a, b). Use the previous exercise to show that f can be defined at a
and b such that f is continuous on (a, b).
10. Suppose that E is a bounded subset of R and f:
ER is uniformly continuous on E. Prove that f is bounded
on E.