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.