Advanced Calculus · Continuity

Continuity is the first property after limits: a limit says where a function tends, continuity says that value must be the one the function takes. Before continuity a function is free to break and jump; after it, differentiation, integration and theorems about maxima and intermediate values have a footing. The course moves in this order: first define continuity at a point, along with left and right continuity, splitting it into being defined, having a limit, and that limit equal to the function value; then study examples of common functions and their breaks; next test continuity, classify discontinuity points, and see that elementary functions are continuous on their domains; and finally use the closed-interval properties, boundedness, the extreme value, intermediate value and zero point theorems, to settle existence questions. By the end you should be able to judge continuity at a point, classify a discontinuity, and prove a root exists.