# Answer to Question #15806 in Real Analysis for ran

Question #15806

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 .

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 .

Expert's answer

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.

2. Yes, from the definition.

3. Yes. Because property of Linearity

4. No.

5. Yes. From theorem about sequence integration.

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