Continuity and Types of Discontinuity in Functions

Continuity and Discontinuity of Functions

Introduction

This study guide covers the concepts of continuity and discontinuity in functions, including definitions, types, and examples. It also introduces the Intermediate Value Theorem, a fundamental result in calculus.

Checking Continuity at a Point

Definition of Continuity at a Point

• A function f(x) is continuous at a point x = a if the following three conditions are satisfied:

  1. f(a) is defined

  2. limx→a f(x) exists

  3. limx→a f(x) = f(a)

• If any of these conditions fail, the function is discontinuous at x = a.

Example: Discontinuity in Piecewise Functions

• Consider $f(x) = \frac{x^2 - x - 2}{x - 2}$ for $x \neq 2$, and $f(x) = 1$ for $x = 2$.

• At $x = 2$:

  • $f(2) = 1$

  • $\lim_{x \to 2} \frac{x^2 - x - 2}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+1)}{x-2} = 2 + 1 = 3$

  • Since $f(2) \neq \lim_{x \to 2} f(x)$, the function is discontinuous at $x = 2$.

Types of Discontinuity

1. Removable Discontinuity

• Definition: Occurs when $f(a)$ is not defined or does not equal $\lim_{x \to a} f(x)$, but $\lim_{x \to a} f(x)$ exists and is finite.

• Graphical Interpretation: There is a 'hole' in the graph at $x = a$.

• Example: $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$ has a removable discontinuity because $\lim_{x \to 1} f(x) = 2$ but $f(1)$ is undefined.

2. Jump Discontinuity

• Definition: Occurs when the left-hand and right-hand limits at $x = a$ exist but are not equal.

• $\lim_{x \to a^-} f(x) = L$, $\lim_{x \to a^+} f(x) = M$, with $L \neq M$.

• Graphical Interpretation: The graph 'jumps' from one value to another at $x = a$.

• Example: The greatest integer function $f(x) = \lfloor x \rfloor$ has jump discontinuities at integer values of $x$.

3. Infinite Discontinuity

• Definition: Occurs when the function approaches infinity as $x$ approaches $a$.

• $\lim_{x \to a} f(x) = \pm \infty$

• Graphical Interpretation: The graph has a vertical asymptote at $x = a$.

• Example: $f(x) = \frac{1}{x - 4}$ has an infinite discontinuity at $x = 4$.

Continuity from the Right and Left

Definitions

• Continuous from the right at $a$: $\lim_{x \to a^+} f(x) = f(a)$

• Continuous from the left at $a$: $\lim_{x \to a^-} f(x) = f(a)$

Example: Piecewise Function

• Given $f(x) = \begin{cases} e^x & x \leq 0 \\ 2 - x & x > 0 \end{cases}$

• At $x = 0$:

  • $f(0) = e^0 = 1$

  • $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} e^x = 1$

  • $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} 2 - x = 2$

  • Since left and right limits are not equal, there is a jump discontinuity at $x = 0$.

  • Function is continuous from the left at $x = 0$.

Continuity on an Interval

Definition

• A function f is continuous on an interval if it is continuous at every number in the interval.

• If defined only on one side at an endpoint, continuity at the endpoint means continuous from the right or left as appropriate.

Making a Piecewise Function Continuous Everywhere

Example: Solving for Parameters

• Given $f(x) = \begin{cases} x^2 - 4 & x < 2 \\ ax^2 - bx + 3 & 2 \leq x \leq 3 \\ 2x - a + b & x > 3 \end{cases}$

• To make $f(x)$ continuous everywhere, solve for $a$ and $b$ so that the function is continuous at $x = 2$ and $x = 3$.

• Set $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x)$ and $\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x)$, then solve the resulting system of equations.

• Example Solution:

  • At $x = 2$: $4a - 2b + 3 = 4$

  • At $x = 3$: $9a - 3b + 3 = 6 - a + b$

  • Solve for $a$ and $b$ to ensure continuity.

The Intermediate Value Theorem

Statement

• The Intermediate Value Theorem: If $f$ is continuous on the closed interval $[a, b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there exists a number $c$ in $(a, b)$ such that $f(c) = N$.

• Application: Used to show the existence of roots or solutions within an interval.

• Example: If $f(a) < 0$ and $f(b) > 0$, then there is some $c$ in $(a, b)$ with $f(c) = 0$.

Summary Table: Types of Discontinuity

Type	Definition	Graphical Feature	Example
Removable	Limit exists, $f(a)$ undefined or not equal to limit	Hole in graph	$f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$
Jump	Left and right limits exist but are not equal	Graph jumps at $x = a$	$f(x) = \lfloor x \rfloor$ at integer $x$
Infinite	Limit approaches $\pm \infty$	Vertical asymptote	$f(x) = \frac{1}{x - 4}$ at $x = 4$

Key Formulas and Equations

• $\lim_{x \to a} f(x)$: The limit of $f(x)$ as $x$ approaches $a$

• $\lim_{x \to a^-} f(x)$: Left-hand limit

• $\lim_{x \to a^+} f(x)$: Right-hand limit

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