Mastering Continuity: Concepts, Theorems, and Analysis

Continuity

The property of a function that allows us to draw its graph without lifting the pencil.

Continuity test

Three conditions must be met for a function to be continuous at a point: 1) f(c) is defined, 2) the limit as x approaches c exists, 3) the limit equals f(c).

Discontinuity

A point at which a function fails to be continuous due to one or more failing conditions of the continuity test.

Removable Discontinuity

A type of discontinuity where the limit exists but the function value is undefined, typically represented as a hole in the graph.

Jump Discontinuity

Occurs when the left-hand limit and right-hand limit are both finite but not equal, resulting in a step in the graph.

Infinite Discontinuity

A type of discontinuity where the function approaches positive or negative infinity as x approaches a point, often associated with vertical asymptotes.

Right-continuous

A function is right-continuous at a point if the limit from the right equals the function's value at that point.

Left-continuous

A function is left-continuous at a point if the limit from the left equals the function's value at that point.

Intermediate Value Theorem (IVT)

States that if a function is continuous on a closed interval [a, b], then for every value between f(a) and f(b), there exists a c in (a, b) such that f(c) equals that value.

Existence Theorem

A theorem that guarantees the existence of a value within a given range without providing a method to find it, specifically the Intermediate Value Theorem.

Non-removable Discontinuity

Includes both jump and infinite discontinuities, where the discontinuity cannot be 'fixed' by redefining the function at a point.

Function Definition

A formal statement that outlines how a function operates and what values it outputs for given inputs.

Polynomials

Functions consisting of variables raised to whole number exponents, which are continuous everywhere in their domain.

Rational Functions

Functions expressed as the quotient of two polynomials, which can have removable or non-removable discontinuities depending on their structure.

Piecewise Function

A function defined by multiple sub-functions, which can exhibit jump discontinuities.

Endpoints

The values at the boundaries of a closed interval, crucial for evaluating continuity across that interval.

Limits

The value that a function approaches as the variable approaches a certain point, integral to determining continuity.

Continuous Function

A function that is uninterrupted and without breaks, holes, or jumps in its graph.

Algebraic Clue

Specific indicators, such as common factors, that help identify types of discontinuities in functions.

Zero (Root) of a Function

A value c for which f(c) = 0; points where a function crosses the x-axis.

Continuous on (a, b)

Describes a function that is continuous at every point within the open interval (a, b).

Continuous on [a, b]

Describes a function that is continuous throughout the closed interval [a, b], including endpoints.

Discontinuity Classification Chart

A table summarizing the existence of limits, whether the function is defined, and the removable status of different types of discontinuities.

Hole in the Graph

A graphical representation of a removable discontinuity, indicating a missing point where the function is undefined.

Function Redefinition

The process of altering the definition of a function at a point to remove a removable discontinuity.

Limit Exists

Indicates that the left-hand and right-hand limits at a point agree on a finite value.

Undefined Function Value

Occurs when a function does not have an output for a given input, often seen at points of discontinuity.