Is h Continuous and Increasing?

  • Thread starter Rosey24
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In summary, the conversation discusses a problem involving a function h with a certain property on a given interval. The problem asks to prove that h is increasing if h is continuous, and to provide a counterexample if h is not continuous. A solution is provided for the second part, and a hint is given for the first part using the intermediate value property.
  • #1
Rosey24
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Homework Statement



We have a worksheet with practice final questions and I'm really stuck on this one on continuity:

Suppose h: (0,1) -> R has the property that for all x in (0,1), there exists a delta>0 such that for all y in (x, x+delta)[tex]\bigcap[/tex](0,1), h(x) <= h(y)

a) prove that if h is continuous on (0,1), then h is increasing.
b) Give a counterexample to show that this need not be true if h is not continuous.

Thanks so much for any help you can provide!

Homework Equations





The Attempt at a Solution





The Attempt at a Solution

 
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  • #2
Think about h(x)=1 for x in (0,1/2] and h(x)=0 in (1/2,1). Open boundaries make all the difference.
 
  • #3
Dick's response is to part (b).

For (a), Suppose u< v in (0, 1). If h(u)> h(v), can you get a contradiction to "there exists a delta>0 such that for all y in (x, x+delta)(0,1), h(x) <= h(y)" using the intermediate value property?
 

Related to Is h Continuous and Increasing?

1. What is a difficult continuity question?

A difficult continuity question is a question that involves understanding and analyzing the continuity of a particular phenomenon or process. It may require critical thinking skills and knowledge of scientific principles to answer.

2. Why are continuity questions considered difficult?

Continuity questions can be difficult because they often involve complex systems or processes that are not easily observed or measured. They may also require a deep understanding of underlying principles and connections between different aspects of the phenomenon.

3. How can one approach a difficult continuity question?

One can approach a difficult continuity question by breaking it down into smaller, more manageable parts and analyzing each part separately. This can help identify patterns and connections that can lead to a better understanding of the phenomenon as a whole.

4. What skills are necessary to successfully answer a difficult continuity question?

To successfully answer a difficult continuity question, one may need skills such as critical thinking, problem-solving, data analysis, and a strong understanding of scientific principles and concepts. Good observation skills and the ability to make connections between different aspects of the phenomenon are also helpful.

5. How can difficult continuity questions contribute to scientific understanding?

Difficult continuity questions can contribute to scientific understanding by challenging scientists to think critically and deeply about a particular phenomenon. By attempting to answer these questions, scientists may uncover new information and connections, leading to a deeper understanding of the natural world and potential applications in various fields.

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