A function f is continuous at c if:
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By applying the process of triple integration (calculus) to the definition of a sphere.
Calculus; by a long shot.
Just about all of calculus is based on differential and integral calculus, including Calculus 1! However, Calculus 1 is more likely to cover differential calculus, with integral calculus soon after. So there really isn't a right answer for this question.
Tangent continuity: No sharp angles. Curvature continuity: No sharp radius changes.
It is certainly used in calculus, just as calculus can be used in trigonometry.