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Criteria for Integrable or differentiable function?

Hello, brilliant users. I've been wondering for a years since I've met calculus. Now I want to ask two questions:

  1. Can we find the criteria for function that only can integrable, but cannot differentiable? What is the example for this function? Why or why not if the function like that doesn't exist?

  2. Can we find the criteria for function that neither integrable nor differentiable? What is/are the example for this function? Thanks.

Note by Leonardo Chandra
3 years, 11 months ago

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Case 1: \(f(x) = |x|\) is continuous everywhere, and so integrable. but is not differentiable at \(0\). You can create continuous functions (which are certainly integrable) which are differentiable at no point, but these are a bit more difficult to find. Have a look on the web for "nowhere differentiable functions", or "Weierstrass function".

Case 2: The function \[ f(x) \; = \; \left\{ \begin{array}{ll} 1 & x \mbox{ rational} \\ 0 & x \mbox{ irrational} \end{array}\right. \] is continuous nowhere, so certainly differentiable nowhere. Whether this function is integrable or not depends on your precise definition of integration. If you work with Riemann integration (the most common sort), then this function is not integrable. There is another theory of integration (Lebesgue integration) for which this function is integrable. However, there are examples of non-differentiable functions which fail to be integrable for that other theory of integration, too.

You might ask why there are a variety of theories of integration. The most intuitive is Riemann integration, which is (basically) the one we start by learning. In very broad terms, it is in the business of approximating areas under a curve by a large number of very narrow vertical strips, so that \[ \int y\,dx \; \approx. \; \sum y \delta x \;.\] The problem with Riemann integration is that it is not easy to prove useful theorems about it, and it is fundamentally a theorem about integration on finite intervals for bounded functions. You need to cheat a little to define an integral over an infinite range, or the integral of an unbounded function, creating what is called the improper Riemann integral.

Lebesgue integration is a theory which, in essence, turns the little strips on their side. It defines an integral by considering functions that can be obtained by piling up horizontal strips instead of vertical ones. Surprisingly, this has a remarkable effect. Firstly, it enables integrals of unbounded functions on bounded intervals to be defined, and is a theory for which really useful theorems can be proven. On the other hand, it is very precise about what being integrable means, and so there is a difference between what it means to be Riemann integrable and Lebesgue integrable. Mark Hennings · 3 years, 11 months ago

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@Mark Hennings For anyone who wants to investigate further, the function stated in case two is the Dirichlet function. Bob Krueger · 3 years, 11 months ago

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Very interesting questions! Unfortunately, my own knowledge in integrals and differentials is too poor to provide an answer to this. However, I would really love to see someone elaborate! Ivan Sekovanić · 3 years, 11 months ago

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