Proving the Intermediate Value Theorem and Range of 1:1 Continuous Functions

In summary, the conversation discusses two problems involving continuous functions. The first problem involves showing that there exists a value of x in the interval [0, 1] where f(x) is equal to f(x+1). The second problem involves showing that the range of a one-to-one and continuous function on an interval [a, b] with f(a) < f(b) is the interval [f(a), f(b)]. Potential solutions are suggested using the Intermediate Value Theorem and the function h(x)=f(x)-f(x+1).
  • #1
busterkomo
1
0

Homework Statement


a)Let f(x) be continuous on [0, 2], with f(0) = f(2). Show that f(x) = f(x+1) for some x ε [0, 1].
b)Let f(x) be 1:1 and continuous on the interval [a, b] with f(a) < f(b). Show that the range of f is the interval [f(a), f(b)].


Homework Equations





The Attempt at a Solution


I'm not really where to start for either of them. In a), I find it obvious that there exists an f(x) = f(x+1) for some x in that interval, but find it difficult to prove without any specific function. I find using the I.V.T. difficult in general without being applied to a specific function. Any help/hints appreciated. Thanks!
 
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  • #2
Hello fellow Waterloo student! I would love to help, but alas, I am having the same problems as you! :S
 
  • #3
Hello busterkomo try taking a function h such that:

h(x)=f(x)-f(x+1)

And h is obviously a continuous function as a difference of two continuous functions.
Now what can you do from that??
 

Related to Proving the Intermediate Value Theorem and Range of 1:1 Continuous Functions

What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a mathematical theorem that states that if a continuous function f(x) is defined on an interval [a, b], and if there are two points, c and d, in that interval such that f(c) is less than or equal to y and f(d) is greater than or equal to y, then there must exist a point x between c and d such that f(x) is equal to y.

How does the Intermediate Value Theorem work?

The Intermediate Value Theorem works by showing that if a continuous function takes on two different values at two points in an interval, then it must also take on every value in between those two points. This is because a continuous function is defined as one where the graph has no breaks or gaps, allowing the function to smoothly connect any two points on the graph.

Why is the Intermediate Value Theorem important?

The Intermediate Value Theorem is important because it provides a way to prove the existence of solutions to equations that cannot be solved algebraically. It also has practical applications in fields such as economics, physics, and engineering, where continuous functions are commonly used to model real-world phenomena.

What are the limitations of the Intermediate Value Theorem?

The Intermediate Value Theorem only applies to continuous functions, which means that there must be no breaks or gaps in the graph of the function. This limits its applicability to certain types of functions that are not continuous, such as piecewise functions or functions with vertical asymptotes. Additionally, the theorem does not provide any information about the location of the point where the function takes on the given value, only that it must exist within the given interval.

How is the Intermediate Value Theorem used in real-world problems?

The Intermediate Value Theorem is commonly used in real-world problems to prove the existence of solutions to equations that cannot be solved algebraically. It can also be used to show that a function has a root or a point of intersection with another function. For example, it can be used to show that there is a time when the velocity of a moving object is equal to a given speed, or that there is a point where the demand and supply curves intersect in economics.

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