Why are continuity and limits considered more basic than derivatives and integrals?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
In the field of mathematical analysis, continuity and limits are considered more fundamental than derivatives and integrals because they provide the rigorous logical foundation upon which the latter are built.1
Historically, while calculus was used successfully for years, it lacked a formal explanation for why its methods worked. In the 19th century, Augustin-Louis, Baron Cauchy established a rigorous basis for the subject by prioritizing continuity and limits. He demonstrated that once these two concepts are precisely defined, the more complex operations of differentiation and integration can be defined in terms of them.1
Specifically:
* Limits allow for the calculation of values where a function is not directly defined and provide the mechanism to define the derivative as a rate of change without relying on vague notions of motion.213
* Continuity describes functions that vary without abrupt breaks, serving as a prerequisite for differentiability; while a differentiable function must be continuous, the reverse is not always true.145
By defining these concepts independently, mathematicians were able to provide a solid, logical underpinning for all of calculus.1