Real Analysis Fall 2004 Take Home Test 1 SOLUTIONS 1. Use the ...

Method 1. Let ε > 0 be given. Let x 0 ∈ R\{0}. Put δ = |x 0|
2 . Then if |x − x 0| < δ = |x 0|
2 ,
then |x 0|
2 < x < 3|x 0|
2 . Let δ 2 = ε
10 |x 0|x 2 0 . For δ = min(δ 1, δ 2 ), if |x − x 0 | < δ, then
1
∣x − 1 2 x 2 ∣ = |x2 0 − x2 |
0 x 2 0x 2
= |x − x 0||x + x 0 |
x 2 0x 2
< |x − x 0| 5 |x 2 0|
x 2 x 2 0
0
4
≤ |x − x 0| · 10
|x 0 |x 2 0
= ε
Method 2. Consider the function g(x) = x 2 . Given ε > 0, let δ 1 = |x 0|
2 . Then |x 0|
2 < x <
3|x 0 |
for all |x−x 0 | < δ 1 . Let δ 2 =
2ε
2
5|x 0 | . Then for δ = min(δ 1, δ 2 ), and x ∈ (x 0 −δ, x 0 +δ),
we have
|x 2 − x 2 0 | = |x − x 0||x + x 0 |
< |x − x 0 | 5|x 0|
2
≤ ε.
Then use a theorem about the reciprocal of a continuous function is continuous.
5. Suppose that f is continuous on [a, b], one-to-one (if x 1 ≠ x 2 , then f(x 1 ) ≠ f(x 2 )), and
f(a) < f(b). Prove that f is monotonically increasing (strictly) on [a, b].
Proof. Suppose that f is not monotonically increasing on [a, b].
b
a
a
b