Assumptions of integration

Question

I have a question regarding integration of a (continuous) function $f(x)\in C[a,b]$. Of course, by the second fundamental theorem of calculus, $$\int_a^b f(x) \ dx=F(a)-F(b)$$ I am wondering if there are any assumptions or circumstances in which this is equivalent to $F(a-b)$?

Thanks in advance.

Answer 1

Yes. If it turns out that $f$ a constant function, then $F$ is an affine map and therefore $F(a)-F(b)=F(a-b)$.

Answer 2

Note that we have

$\int_b^a f(x)dx=F(a)-F(b)=F(a-b)\iff F$ is linear that is $F=kx\implies f=k$