Proving Existence of x for Continuous Identity Function f[a,b]->[a,b]

In summary, the intermediate value theorem states that for a continuous function f[a,b]->[a,b], there exists at least one x in [a,b] such that f(x)=x. This can be proven by considering the identity function g(x)=x and using the fact that the graph of f must intersect g at some point. By setting g(x)=f(x)-x, we can see that f(a)-a and f(b)-b are both in [a,b], thus satisfying the conditions of the theorem.
  • #1
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"Let f[a,b]->[a,b] be continuous. Prove that there exists at least 1 x in [a,b] such that f(x)=x."

This seems simple geometrically since if we consider the identity function g(x)=x, if f(x) is continuous, then if you "draw" the graw of f, it must intersect g at some point. At that point, f(x)=x. But I have no idea how to translate this intuition into analytical lingo.
 
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  • #2
Use the intermediate value theorem.
 
  • #3
and what can you say about g(x) = f(x)-x on [a,b]?
 
  • #4
In particular, what is f(a)- a? What is f(b)-b?
(Remember that f(a) and f(b) is in [a,b]?)
 
  • #5
I got it! Thanks for the help.
 

FAQ: Proving Existence of x for Continuous Identity Function f[a,b]->[a,b]

What is a continuous identity function?

A continuous identity function is a type of mathematical function that maps every element in a given interval [a,b] to itself. In other words, the output of the function is always equal to its input. This type of function is represented by the equation f(x) = x and is considered a special case of a continuous function.

What does it mean to prove existence of x for a continuous identity function?

To prove existence of x for a continuous identity function means to show that for any input value within the given interval [a,b], there exists an output value that is equal to the input value. In other words, the function is defined and continuous for all values in the interval, and there are no gaps or missing values in its output.

How is the existence of x for a continuous identity function proven?

The existence of x for a continuous identity function can be proven using various methods, such as the intermediate value theorem or the Bolzano-Weierstrass theorem. These theorems use mathematical principles and properties to demonstrate that the function is continuous and has an output for every input within the given interval.

What is the significance of proving existence of x for a continuous identity function?

Proving existence of x for a continuous identity function is important in mathematics because it confirms that the function is well-defined and continuous, which is essential for many mathematical concepts and applications. It also allows for the use of other mathematical principles and theorems that rely on the function being continuous.

Can the existence of x for a continuous identity function be disproven?

No, the existence of x for a continuous identity function cannot be disproven. This is because the function is defined for all values in the given interval and follows the simple rule of mapping every input to itself. As long as the function is well-defined and continuous, its existence cannot be disproven.

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