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.


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).
