Answer to Question #15806 in Real Analysis for ran

Question

Which of the following statements are true and why?

1.Any continuous function from the open unit interval (0,1) to itself has a fixed point.

2.logx is uniformly continuous on (1/2,+∞) .

3.If A,B are closed subsets of [0,∞) , then A+B={x+y|x∈A,y∈B} is closed in [0,∞)
4.A bounded continuous function on R is uniformly continuous.

5.Suppose f n (x) is a sequence of continuous functions on the closed interval [0,1] converging to 0 pointwise. Then the integral ∫ 1 0 f n (x)dx converges to 0 .

Answer 1

1.No. Fixed point is f(x)=x. Any words about "open unit intervals"

2. Yes, from the definition.

3. Yes. Because property of Linearity

4. No.

5. Yes. From theorem about sequence integration.