@TheGreatDuck Sorry, but I do not know wht the sentence mean.
Are you referring to this? "The indefinite integral of a function MUST be continuous (or fixably discontinuous) for the general scope of the function."
I came up with these few theorems and I am curious whether or not my hypothesis are true. I don't really have a method for proving these, mind you. I'm more considering these as things I've noticed to be true. I just don't know whether they are always true.
Jump Series Theorem
For any function ...
One can prove the following
THM Let $f:[a,b]\to\Bbb R$ be Riemann integrable over its domain. Define a new function $F:[a,b]\to\Bbb R$ by $$F(x)=\int_a^x f(t)dt$$ Then $F$ is continuous. That is, the map $$f\mapsto \int_a^x f$$ sends $\mathscr R[a,b]$ to $\mathscr C[a,b]$.
PROOF Let $c\in[a...
No, the continuing continuity one. The one that states removing the jumps is unnecessary for subsequent integrals. It /seems/ like that step in the process would be unneeded if the previous one is continuous but either my sense of things is wrong or that statement is false.
One can prove the following
THM Let $f:[a,b]\to\Bbb R$ be Riemann integrable over its domain. Define a new function $F:[a,b]\to\Bbb R$ by $$F(x)=\int_a^x f(t)dt$$ Then $F$ is continuous. That is, the map $$f\mapsto \int_a^x f$$ sends $\mathscr R[a,b]$ to $\mathscr C[a,b]$.
PROOF Let $c\in[a...
