Continuity: Definitions, Examples, and Interval Analysis 2.6

2.6 Continuity

Introduction to Continuity

Continuity is a fundamental concept in calculus, describing the behavior of functions at specific points and over intervals. A function is continuous at a point if its value and its limit at that point are equal. Discontinuities occur where this condition fails.

Discontinuities of Functions

Discontinuities are points where a function is not continuous. These can be identified visually on a graph or analytically by examining the function's definition.

• Types of Discontinuities: Removable, jump, and infinite discontinuities.

• Example: For a function $f(x)$, discontinuities may occur at $x = 1, 2, 3, 5$ as shown in a sample graph.

Continuity Checklist at a Point

To determine if a function $f(x)$ is continuous at $x = a$, use the following checklist:

• Existence of $f(a)$: The function must be defined at $a$.

• Existence of the limit: $\lim_{x \to a} f(x)$ must exist.

• Equality: $\lim_{x \to a} f(x) = f(a)$

Formula:

$\lim_{x \to a} f(x) = f(a)$

Examples: Continuity at Specific Points

• Example 1: $f(x) = \frac{3x^2 + 2x + 1}{x - 1}$

  • At $x = 1$: $f(1)$ does not exist (DNE), so not continuous at $x = 1$.

  • At $x = 2$: $f(2) = 17$, $\lim_{x \to 2} f(x) = 17$, so continuous at $x = 2$.

• Example 2: $f(x) = \begin{cases} \frac{x^2 - 36}{x - 6} & x \neq 6 \\ 7 & x = 6 \end{cases}$

  • $f(6) = 7$

  • $\lim_{x \to 6} f(x) = \lim_{x \to 6} \frac{(x+6)(x-6)}{x-6} = \lim_{x \to 6} (x+6) = 12$

  • Since $12 \neq 7$, not continuous at $x = 6$.

• Example 3: $f(x) = \begin{cases} x^2 + 1 & x \leq 0 \\ 3x + 5 & x > 0 \end{cases}$

  • $f(0) = 1$

  • $\lim_{x \to 0^-} f(x) = 1$

  • $\lim_{x \to 0^+} f(x) = 5$

  • Since left and right limits are not equal, not continuous at $x = 0$.

Continuity at Endpoints

For functions defined on closed intervals $[a, b]$, continuity at endpoints is defined as follows:

• Right-continuous at $a$: $\lim_{x \to a^+} f(x) = f(a)$

• Left-continuous at $b$: $\lim_{x \to b^-} f(x) = f(b)$

Continuity on an Interval

To determine where a function is continuous, analyze its domain and points of discontinuity.

• Example 1: $f(x) = \frac{x^2 - 5x + 6}{x^2 - 4}$

  • Denominator zero at $x = 2, -2$

  • Continuous on $(-\infty, -2)$, $(-2, 2)$, $(2, \infty)$

• Example 2: $f(x) = \sqrt{x - 4}$

  • Domain: $x \geq 4$

  • Continuous on $[4, \infty)$

• Example 3: $f(x) = \begin{cases} x^2 + 3x & x > 2 \\ 4x & x \leq 2 \end{cases}$

  • $f(2) = 8$

  • $\lim_{x \to 2^-} f(x) = 8$

  • $\lim_{x \to 2^+} f(x) = 10$

  • Not continuous at $x = 2$; continuous on $(-\infty, 2)$ and $(2, \infty)$

Summary Table: Continuity Conditions

Condition	Description
Function defined at $a$	$f(a)$ exists
Limit exists at $a$	$\lim_{x \to a} f(x)$ exists
Limit equals function value	$\lim_{x \to a} f(x) = f(a)$

Additional info:

• Continuity is essential for defining derivatives and integrals in calculus.

• Piecewise functions often require checking continuity at the points where the formula changes.