In fact, the situation concerning an antiderivative F to a given
f on [a,b] is not as simple. The most
natural definition would be different: We would ask that
F ′ = f on
(a,b) and also that the derivative from the right
of F at a be equal to f (a), while
the derivative from the left of F at b be equal to
f (b).
While this looks like the right definition, it has a serious drawback. It is
too strong for our purposes (it requires more of F than we really need
in applications), which means that many functions that would otherwise work
are ruled out in this way.
Since the existence of one-sided derivatives would imply that such a function
F is continuous on [a,b] (see
Derivative in Derivatives
- Theory - Introduction), it follows that the definition
using continuity that we adopted instead is weaker. This means that when we
adopt the continuity definition instead of the "one-sided derivative
type",
we increase the chance that an antiderivative can be found. And since it
turns out that the continuity condition is exactly what is needed in
applications, it was decided to do it this way.