Are discontinuous functions integrable? And integral of every continuous function continuous?
According to me answer of second part is yes as integration simply means area under curve.
$\endgroup$4 Answers
$\begingroup$Is every discontinuous function integrable?
No. For example, consider a function that is $1$ on every rational point, and $0$ on every irrational point. What is the integral of this function from $0$ to $1$? It's not integrable! For any partition of $[0,1]$, every subinterval will have parts of the function at height $0$ and at height $1$, so there' no way to make the Riemann sums converge. (However you might later encounter something called Lebesgue integration, where they would say this is integrable. 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).
Is the integral of every continuous function continuous?
Yes! In fact, this is a byproduct of what's commonly known as the second fundamental theorem of calculus (although logically it comes first).
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ $\begingroup$First question: yes some are of course (e.g. step functions). Second: yes in fact Lipschitz continuous
$\endgroup$ 2 $\begingroup$Every continuous function is integrable, but some discuntinuous functions are integrable and others aren't.
Moreover, there are two widely-used senses for the word 'integrable': Riemann's integrability and Lebesgue's integrability.
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