Question about Proof of Inverse Function

In summary, the proof of the Inverse Function Theorem in baby Rudin uses the concept of a fixed point and the contraction principle to show that for every point y, the associated function φ must be one-to-one. This is achieved by first showing that |φ'(x)| < ½ and then using theorem 9.19, which states that for a contraction mapping with M = ½, the associated function φ must be one-to-one. This concept can also be applied to the equation f(x) = x², where A⁻¹ represents the inverse of A.
  • #1
transphenomen
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I am reading the proof of the Inverse Function Theorem in baby Rudin and I have a question about it. How does associating a function phi(x) (equation 48) with each point y tell us anything about if f(x) is one-to-one? I'll show the proof below. Also, if f'(a) = A, and f(x)=x2, what would A-1 be?
 

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  • #2
The exact same equation had me freaking out for a while.

You can think of φ(x) = x + A⁻¹(y - f(x)) as coming from f(x) = f(x₀) + f'(x)(x - x₀),
& it's a way of introducing the concept of a fixed point so we can use the contraction
principle. You basically want to show that |φ'(x)| < ½ & then use theorem 9.19
|φ(b) - φ(a)| ≤ M(b - a) & with M = ½ you have a contraction mapping, thus by the
contraction principle you have that for every y it's associated function φ must be 1-1.
 

1. What is the proof of inverse function theorem?

The inverse function theorem states that if a function is differentiable at a point, then its inverse is also differentiable at the corresponding point, with a derivative that is the reciprocal of the original function's derivative.

2. Why is the proof of inverse function important?

The proof of inverse function is important because it allows us to determine whether a function has an inverse and if so, what properties the inverse function has.

3. How do you prove the inverse function theorem?

To prove the inverse function theorem, we first need to show that the function is one-to-one and onto. Then, we can use the derivative of the function to show that the inverse function is also differentiable at the corresponding point.

4. Can the proof of inverse function be applied to any function?

No, the proof of inverse function can only be applied to differentiable functions. If a function is not differentiable, the inverse function theorem does not apply.

5. What are the practical applications of the proof of inverse function?

The proof of inverse function has many practical applications, such as in optimization problems, solving systems of equations, and in calculus to find the inverse of a function to evaluate integrals. It is also used in fields such as physics, engineering, and economics to model and analyze real-world problems.

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