Can you integrate a non continuous function?

Discontinuous functions can be integrable, although not all are. Specifically, for Riemann integration (our normal basic notion of integrals) a function must be bounded and defined everywhere on the range of integration and the set of discontinuities on that range must have Lebesgue measure zero.

Can you take the antiderivative of a discontinuous function?

Therefore, there is no such thing as an antiderivative of a discontinuous function, because that would not be differentiable. Let’s take a closer look at finding a formula for the area under the graph of the function f(t) = {t} from say 0 to x, where x is some real number.

What are some examples of non continuous integrable functions?

You can use these theorems to give examples of noncontinuous integrable functions. By 1 and 3, any function that’s continuous except at finitely many places is integrable. For example, the signum function is not continuous at 0, but it’s integrable.

How do you know if a function is integrable?

By 1 and 3, any function that’s continuous except at finitely many places is integrable. For example, the signum function if the interval of integration is the finite union of intervals such that on each of the subintervals the function is integrable, then the function is integrable on the entire interval.

What is an example of a non-Lebesgue integrable function?

Giving an explicit example of a non-Lebesgue integrable function is harder and more annoying. A good heuristic for such a function would be a function that is 1 at every rational, and a random number between − 1 and 1 for every irrational point – somehow every more discontinuous than the previous example).

Are all functions integrable if they have measure zero?

There is a theorem that says that a function is integrable if and only if the set of discontinuous points has “measure zero”, meaning they can be covered with a collection of intervals of arbitrarily small total length. First question: yes some are of course (e.g. step functions).