Continuity in Calculus: Definitions, Types, and Applications

Section 2.5: Continuity

Definition of Continuity at a Point

Continuity at a point is a fundamental concept in calculus, ensuring that a function behaves predictably without sudden jumps or breaks at a specific value.

• Definition: A function f is continuous at a point c if the following three conditions are met:

  1. f(c) is defined: The function has a value at c.

  2. \( \lim_{x \to c} f(x) \) exists: The limit of f(x) as x approaches c exists.

  3. \( \lim_{x \to c} f(x) = f(c) \): The value of the function at c equals the limit as x approaches c.

• If any of these conditions fail, f is discontinuous at c.

Example: For f(x) to be continuous at x = 2, all three conditions above must be satisfied at x = 2.

Types of Discontinuities

Discontinuities occur when a function is not continuous at a point. The main types are:

Type	Description	Graphical Feature
Removable Discontinuity	Occurs when the limit exists, but f(c) is either undefined or not equal to the limit.	"Hole" in the graph at c
Infinite Discontinuity	Occurs when the function approaches infinity near c.	Vertical asymptote at c
Jump Discontinuity	Occurs when the left and right limits at c exist but are not equal.	"Jump" in the graph at c

Identifying Discontinuities from a Graph

To determine where a function is not continuous, examine the graph for breaks, holes, or jumps.

• Jump Discontinuity: At x = 1 (the graph jumps to a new value).

• Removable Discontinuity: At x = 2 and x = 4 (holes in the graph).

Example: If you must lift your pencil to continue drawing the graph, the function is not continuous at that point.

One-Sided Continuity

Continuity can be considered from the left or right at endpoints or points of interest.

• Left Continuity at c: \( \lim_{x \to c^-} f(x) = f(c) \)

• Right Continuity at c: \( \lim_{x \to c^+} f(x) = f(c) \)

• At x = 1: right continuous

• At x = 2 and x = 4: neither left nor right continuous

Classes of Continuous Functions

Certain types of functions are continuous everywhere in their domains:

• Polynomial functions

• Rational functions (except where the denominator is zero)

• Radical (root) functions (where the radicand is defined)

• Trigonometric and inverse trigonometric functions (in their domains)

• Exponential and logarithmic functions (in their domains)

• Compositions of continuous functions are continuous on the intersection of their domains.

Determining Continuity of a Function

To determine if a function is continuous at a point, use the three-step definition and properties of limits.

• Example: For \( f(x) = \frac{x^2 - x - 6}{x - 3} \) if \( x \neq 3 \), and \( f(x) = 2x - 5 \) if \( x = 3 \):

• Check if \( \lim_{x \to 3} f(x) \) exists and equals \( f(3) \).

• Here, \( \lim_{x \to 3} \frac{(x+2)(x-3)}{x-3} = 3 + 2 = 5 \), but \( f(3) = 1 \), so the function is not continuous at x = 3.

Piecewise Functions and Continuity

For piecewise functions, check continuity at the boundaries where the formula changes.

• Example: \( f(x) = x^3 - bx \) if \( x < 2 \), \( f(x) = 8 - 2b \) if \( x = 2 \).

• Set the left and right limits equal at x = 2 to solve for b:

$\lim_{x \to 2^-} f(x) = 2^3 - b \cdot 2 = 8 - 2b$ $\lim_{x \to 2^+} f(x) = 8 - 2b$ Set equal: $8 - 2b = 8 - 2b$ (always true), so f is continuous for all b.

Interval Notation for Continuity

Describe where a function is continuous using interval notation, excluding points of discontinuity.

• Example: If a function is discontinuous at x = 3, it is continuous on \(( -\infty, 3 ) \cup ( 3, \infty )\).

Summary Table: Types of Discontinuities

Discontinuity Type	Limit Exists?	Function Value Exists?	Limit Equals Function Value?
Removable	Yes	No or Yes (but not equal to limit)	No
Jump	No (left and right limits not equal)	Yes	No
Infinite	No (limit is infinite)	May or may not exist	No

Key Formulas and Notation

• Limit Definition: $\lim_{x \to c} f(x)$

• Continuity at a Point: $\lim_{x \to c} f(x) = f(c)$

• One-Sided Limits: $\lim_{x \to c^-} f(x)$ (from the left), $\lim_{x \to c^+} f(x)$ (from the right)

Applications

• Continuity is essential for defining derivatives and integrals.

• Many physical phenomena (e.g., motion, growth) are modeled by continuous functions.

Additional info: Some examples and explanations were expanded for clarity and completeness based on standard calculus curriculum.