Last part of question on continuous functions

You can use the intermediate value theorem to prove this.In summary, the conversation discusses a revision question involving a function h: (0,1) -> satisfying certain conditions and being continuous on the interval. The task is to prove that h(x) <= h(y) whenever x and y are in the interval and x <= y. The intermediate value theorem is suggested as a possible approach, but the person is struggling to see how to use it. They are seeking help and any pointers would be appreciated.
  • #1
nlews
11
0

Homework Statement


This is the last part of a revision question I'm trying, would really like to get to the end so any pointers or help would be greatly appreciated.

Suppose h:(0,1)-> satisfies the following conditions:
for all xЭ(0,1) there exists d>0 s.t. for all x'Э(x, x+d)n(0,1) we have h(x)<=h(x')

Prove that if h is continuous on (0,1) then h(x)<=h(y) whenever x.yЭ(0,1) and x<=y. Use a counterexample to show that this results may not be true when h is continuous.

Homework Equations


Well, definition of continuous functions,
also above in the question i am asked to state the intermediate value theorem so i think perhaps i am meant to use that.


The Attempt at a Solution



if h is continuous on (0,1) then for all cЭ(0,1), for all E>0, Эd s.t. for all xЭ(0,1), 0<|x-c|<d this implies that |h(x)-h(c)|<E
i really just don't see how to continue. Any help would be great. Thank you
 
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  • #2
Try fixing x, and showing that for every value of y larger than it, h(x)<=h(y)
 

FAQ: Last part of question on continuous functions

What is the definition of a continuous function?

A continuous function is a mathematical function that maintains its value throughout its domain, meaning that there are no abrupt changes or jumps in the function. This means that as the input values of the function get closer together, the output values also get closer together.

How do you determine if a function is continuous at a specific point?

A function is considered continuous at a specific point if the limit of the function at that point exists and is equal to the value of the function at that point. This means that the left and right limits of the function at that point must be equal.

Can a function be continuous at one point but not at others?

Yes, it is possible for a function to be continuous at one point but not at others. For example, a function may be continuous at all points except for a single point where there is a removable discontinuity.

What is the difference between a continuous function and a differentiable function?

A continuous function is a function that maintains its value throughout its domain, while a differentiable function is one that has a defined derivative at all points in its domain. This means that a differentiable function is also continuous, but the converse is not always true.

What are some real-life applications of continuous functions?

Continuous functions have many real-life applications in fields such as physics, engineering, and economics. For example, they can be used to model the motion of objects, the growth of populations, and the behavior of stock prices. They are also used in calculus to solve optimization problems and find areas under curves.

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