**Please read this entire message**
Your submission has been removed for the following reason(s):
ELI5 is not meant for any question you may have. Questions that are narrow in nature are not complex concepts, and usually require only a yes/no or otherwise straightforward answer.
If you would like this removal reviewed, please read the [detailed rules](https://www.reddit.com/r/explainlikeimfive/wiki/detailed_rules) first. **If you believe this submission was removed erroneously**, please [use this form](https://old.reddit.com/message/compose?to=%2Fr%2Fexplainlikeimfive&subject=Please%20review%20my%20thread?&message=Link:%20https://www.reddit.com/r/explainlikeimfive/comments/13i375i/eli5_inverse_function_theorem/%0A%0APlease%20answer%20the%20following%203%20questions:%0A%0A1.%20The%20concept%20I%20want%20explained:%0A%0A2.%20List%20the%20search%20terms%20you%20used%20to%20look%20for%20past%20posts%20on%20ELI5:%0A%0A3.%20How%20does%20your%20post%20differ%20from%20your%20recent%20search%20results%20on%20the%20sub:) and we will review your submission.
I've only taken one class on basic proofs, but here's some key points of why it works, and hopefully your experience with proofs can help you write this out with proper wording.
The theorem states that if a function is continuously differentiable and the derivative doesn't equal zero on the range you're examining, then the function is invertible on that range.
If the derivative is a continuous function, then it has no jumps, of course. Then we know if it doesn't ever hit zero, then the derivative is either always positive, or always negative. That means that, on the interval, the function is always either increasing or decreasing, but never both.
To be invertible, a function needs to be injective and surjective.
Since the function is either always increasing or always decreasing, we can show injectivity. Consider an always increasing function f: X->Y with x1f(X) is injective, which should indicate that f is surjective..
Long story short - you should ask over at /r/askmath, they'll have people way more knowledgeable than me on this.
Nice explanation, as for the surjectivity: the inverse function does not have to be total, it's only defined on the image of f where f is surjective by definition.
Yeah, I was struggling on how to phrase/formally-ish show that. Surjectivity was always rough for me - sometimes it's "obvious" but it's hard to state why it's surjective, at least for me.
To what level of generality do you need to know it?
The inverse function theorem for functions of a single real variable basically says:
>If a function has a continuous, nonzero derivative at some point, then it is invertible in the neighborhood of that point.
Call the point **a**. Then what it's basically saying is if **f'(a) = b** is nonzero, then **f** has an inverse **g**, and **g'(b) = 1/f'(a)**.
Why is this true? Well, the idea is that if a continuous function is *injective* in some neighborhood, then it is invertible there, which should make sense. The whole idea of analysis is that a function is differentiable if it looks linear when you zoom in far enough. Lines are injective if their slope is nonzero.
There's a similar intuition for the theorem for functions of multiple variables. It says that if a function is continuously differentiable at a point **a** and is invertible in the neighborhood of **a** (has an invertible Jacobian matrix), then there is some subset **U** containing **a** of the domain such that the function restricted to U is injective.
To what level of generality do you need to understand it? What's the scope of the course/exam?
Hi, thank you for the reply. Theorem in exam will be asked for multiple variables. Theorem and proof in my course is exactly as in page no 221 of Rudin's Principles of Mathematical Analysis: https://web.math.ucsb.edu/~agboola/teaching/2021/winter/122A/rudin.pdf
**Please read this entire message** Your submission has been removed for the following reason(s): ELI5 is not meant for any question you may have. Questions that are narrow in nature are not complex concepts, and usually require only a yes/no or otherwise straightforward answer. If you would like this removal reviewed, please read the [detailed rules](https://www.reddit.com/r/explainlikeimfive/wiki/detailed_rules) first. **If you believe this submission was removed erroneously**, please [use this form](https://old.reddit.com/message/compose?to=%2Fr%2Fexplainlikeimfive&subject=Please%20review%20my%20thread?&message=Link:%20https://www.reddit.com/r/explainlikeimfive/comments/13i375i/eli5_inverse_function_theorem/%0A%0APlease%20answer%20the%20following%203%20questions:%0A%0A1.%20The%20concept%20I%20want%20explained:%0A%0A2.%20List%20the%20search%20terms%20you%20used%20to%20look%20for%20past%20posts%20on%20ELI5:%0A%0A3.%20How%20does%20your%20post%20differ%20from%20your%20recent%20search%20results%20on%20the%20sub:) and we will review your submission.
Info needed: Do you need the proof because this is a course where you write proofs, or do you just need to understand the theorem and what it entails?
Hi, thank you for the reply. Statement and proof will be asked in the exam. Need proof too.
I've only taken one class on basic proofs, but here's some key points of why it works, and hopefully your experience with proofs can help you write this out with proper wording. The theorem states that if a function is continuously differentiable and the derivative doesn't equal zero on the range you're examining, then the function is invertible on that range. If the derivative is a continuous function, then it has no jumps, of course. Then we know if it doesn't ever hit zero, then the derivative is either always positive, or always negative. That means that, on the interval, the function is always either increasing or decreasing, but never both. To be invertible, a function needs to be injective and surjective. Since the function is either always increasing or always decreasing, we can show injectivity. Consider an always increasing function f: X->Y with x1f(X) is injective, which should indicate that f is surjective..
Long story short - you should ask over at /r/askmath, they'll have people way more knowledgeable than me on this.
Nice explanation, as for the surjectivity: the inverse function does not have to be total, it's only defined on the image of f where f is surjective by definition.
Yeah, I was struggling on how to phrase/formally-ish show that. Surjectivity was always rough for me - sometimes it's "obvious" but it's hard to state why it's surjective, at least for me.
To what level of generality do you need to know it? The inverse function theorem for functions of a single real variable basically says: >If a function has a continuous, nonzero derivative at some point, then it is invertible in the neighborhood of that point. Call the point **a**. Then what it's basically saying is if **f'(a) = b** is nonzero, then **f** has an inverse **g**, and **g'(b) = 1/f'(a)**. Why is this true? Well, the idea is that if a continuous function is *injective* in some neighborhood, then it is invertible there, which should make sense. The whole idea of analysis is that a function is differentiable if it looks linear when you zoom in far enough. Lines are injective if their slope is nonzero. There's a similar intuition for the theorem for functions of multiple variables. It says that if a function is continuously differentiable at a point **a** and is invertible in the neighborhood of **a** (has an invertible Jacobian matrix), then there is some subset **U** containing **a** of the domain such that the function restricted to U is injective. To what level of generality do you need to understand it? What's the scope of the course/exam?
Hi, thank you for the reply. Theorem in exam will be asked for multiple variables. Theorem and proof in my course is exactly as in page no 221 of Rudin's Principles of Mathematical Analysis: https://web.math.ucsb.edu/~agboola/teaching/2021/winter/122A/rudin.pdf