Continuity Basic Introduction, Point, Infinite, & Jump Discontinuity, Removable & Nonremovable

Study Guide: Continuity

Definition of Continuity: Continuity refers to the property of a function that indicates it does not have any 'jumps' or 'breaks' in its graph. Mathematically, a function f is continuous at a point x = a if:

f(a) is defined,
(\lim_{x \to a} f(x)) exists, and
(\lim_{x \to a} f(x) = f(a)).

Types of Continuity:

Point Continuity: A function is continuous at a single point if it meets the three conditions mentioned above.
Interval Continuity: A function is continuous over an interval if it is continuous at every point within that interval.
Uniform Continuity: A stronger form of continuity where a function is continuous such that the rate of change of the function does not vary too much over its domain.

Importance of Continuity:

Continuity is crucial in calculus because it ensures the existence of derivatives.
Continuous functions can be integrated easily, and many fundamental theorems in calculus rely on the continuity of functions.
Ensures predictable behavior of functions in mathematical modeling and real-world applications.

Examples:

Continuous functions: polynomials, sine and cosine functions.
Functions that are not continuous: piecewise functions with jumps, functions with asymptotes.

Applications:

Used in real-life scenarios such as physics (motion), economics (cost functions), and engineering (design and analysis).

Concepts Related to Continuity:

Intermediate Value Theorem: If a function is continuous on a closed interval [a, b], then it takes every value between f(a) and f(b).
Extreme Value Theorem: If a function is continuous on a closed interval [a, b], it has both a maximum and minimum value on that interval.