Continuity is the property that makes functions “well behaved”: intuitively, a graph that can be traced without lifting the pen from the paper. The chapter defines it precisely, starting from the concept of limit (three simultaneous conditions), shows that the elementary functions and their combinations are continuous on their domain, and studies how to check continuity at the critical points of piecewise-defined functions. When continuity fails, discontinuity is classified into three types — first kind (jump), second kind (infinite or non-existent limit) and third kind (removable). On closed and bounded intervals, continuity guarantees powerful global results: Weierstrass’s theorem (existence of an absolute maximum and minimum), the intermediate value theorem and the zero theorem. From the last of these arise the numerical methods for solving equations: the bisection algorithm and the fast Newton-Raphson method.
Sezioni
- Definition of continuity
- Continuity of piecewise functions
- Types of discontinuity
- Theorems on continuous functions
- Bisection algorithm and Newton’s method