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I’m working on a mathematics writing question and need an explanation to help me learn.

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).

EXERCISES 4.3

1. Prove Theorem 4.3.3.

2. Show that the following functions are not uniformly continuous on the given domain.

1

*a. f(x) = *. Dom f = (0.00)

b. 8(x) Dom g =

c. h(x) = sin Dom h =

(0,00)

x2

(0.00)

3. Prove that each of the following functions is uniformly continuous on the indicated set.

*a. f(x)

x€ [0, …)

b. 8(x) = r?, XEN

.

1 + x

1

c. h(x)

XER d. k(x) = cosx, ER

r? +1′

r

sinx

e. e(x)

x€ (0,00) *l. f(x) = x€ (0, 1)

x + 1

4. Show that each of the following functions is a Lipschitz function.

1

*a. f(x) Dom f = (a,00), a > 0 b. 8(x)

–

X

x

Dom 8 = (0,00)

+1′

ch(x) = sin Dom h = = (a,00), a > 0 d. p(x) a polynomial, Dom p = = (-a, a), a > 0

5. *a. Show that f(x) = Vx satisfies a Lipschitz condition on (a,00), a > 0.

b. Prove that Vx is uniformly continuous on (0.00).

c. Show that f does not satisfy a Lipschitz condition on (0,00).

6. Suppose E C R 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 E C R 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, f 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 E CR and f: E-R is uniformly continuous. If {x,} 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: E-R is uniformly continuous on E. Prove that f is bounded

on E.

4.3

Uniform Continuity

In the previous section we discussed continuity of a function at a point and on a set. By

Definition 4.2.1, a function f:E → R is continuous on E if for each p E E, given any

€ > 0, there exists a 8 > O such that \(x) – f(p)] 0, the choice of 8 that works depends not only on e and the func-

tion f, but also on the point p. This was illustrated in Example 4.1.2(g) for the function

f(x) = 1/x.x € (0,00). Functions for which a choice of 8 independent of p is possible

are given a special name.

4.3.1 DEFINITION Lei ECR and f: E™R. The function is uniformly continuous

on E if given e > 0, there exists a 8 >0 such that

\f(x) – f(y)] 0 be given. Take S = €/2C. If x, y E E with lx – ył 0 such that

Vix) = f()) = 1*-} 0. Suppose x, y E (a,). Then

THEOREM If KCR is compact and f: K+R is continuous on K, then f is uni-

formly continuous on K.

Proof. Let e > 0 be given. Since f is continuous, for each pe K, there exists a

Op > O such that

\F(x) – f() 0, if we choose & such that 0 0, are examples of an extensive class of functions. If E CR, a

function f: EnR satisfies a Lipschitz condition on E if there exists a positive con-

stant M such that

\f(x) – f(y)] = Mpx – y

for all x, y E E. Functions satisfying the above inequality are usually referred to as Lip-

schitz functions. As we will see in the next chapter, functions for which the derivative

is bounded are Lipschitz functions. As a consequence of the following theorem, every

Lipschitz function is uniformly continuous. However, not every uniformly continuous

function is a Lipschitz function. For example, the function f(x) = Vr is uniformly

continuous on (0,00), but f does not satisfy a Lipschitz condition on [0, 0) (see Ex-

ercise 5).

8 = min{de, : i = 1,…,n}.

Then 8 > 0. Suppose x, y E K with [x – yl

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