In my class we have said that an integral from -infinity to infinity of a function f exists, when a point c exists, such that both improper integrals (Integral from -infinity to c of f) and (Integral from c to infinity) exist.
My question is: we say that at least one point c should exist, but doesn't the existence of such point imply that all integrals with a random point x from -infinity to x and from x to infinity exist, or else such point c would not have existed since we have established with c that we can integrate over the whole area of the function f?