2.3 Continuity and Types of Discontinuities

Continuity is a key concept in calculus that describes how functions behave at specific points. It's all about smooth, unbroken curves without gaps or jumps. Understanding continuity helps us analyze function behavior and predict outcomes in real-world scenarios.

We'll look at different types of discontinuities, like removable points and jumps. These are crucial for spotting where functions break down or change abruptly. Knowing these helps us solve problems and understand function limitations in practical applications.

Continuity and Continuous Functions

Defining Continuity

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• Continuity is a fundamental concept in calculus that describes the behavior of a function at a particular point or over an interval
• A function $f(x)$ is continuous at a point $a$ if the following three conditions are satisfied:
  1. $f(a)$ is defined (the function exists at $a$)
  2. $\lim_{x \to a} f(x)$ exists (the limit of the function as $x$ approaches $a$ exists)
  3. $\lim_{x \to a} f(x) = f(a)$ (the limit of the function as $x$ approaches $a$ is equal to the value of the function at $a$)
• Informally, a continuous function is one whose graph can be drawn without lifting the pencil from the paper

Continuous Functions

• A continuous function is a function that is continuous at every point in its domain
• If a function is continuous on a closed interval $[a, b]$, it satisfies the following properties:
  1. The function is defined at every point in the interval $[a, b]$
  2. The function has no breaks, gaps, or holes in its graph over the interval $[a, b]$
  3. The function has no vertical asymptotes in the interval $[a, b]$
• Examples of continuous functions include polynomials ($f(x) = x^2 + 3x - 1$), exponential functions ($f(x) = e^x$), and trigonometric functions ($f(x) = \sin(x)$) on their domains

Piecewise Functions and Continuity

• A piecewise function is a function that is defined by different formulas for different intervals in its domain
• To determine if a piecewise function is continuous, check the continuity at each endpoint of the intervals where the function changes its definition
• For a piecewise function to be continuous, it must satisfy the following conditions:
  1. Each piece of the function must be continuous on its respective interval
  2. At the endpoints where the function changes its definition, the limit of the function from the left must equal the limit of the function from the right (left-hand and right-hand limits must agree)
  3. At the endpoints where the function changes its definition, the common limit must equal the value of the function at that point
• Example: Consider the piecewise function $f(x) = \begin{cases} x^2 & \text{for } x < 1 \\ 2x - 1 & \text{for } x \geq 1 \end{cases}$. To determine if $f(x)$ is continuous, check the continuity at $x = 1$ by evaluating the left-hand and right-hand limits and comparing them to the value of the function at $x = 1$

Types of Discontinuities

Point Discontinuity

• A point discontinuity occurs when a function is discontinuous at a single isolated point
• There are two types of point discontinuities:
  1. Removable discontinuity: The limit of the function exists at the point, but the function is either undefined or has a different value at that point
  2. Jump discontinuity: The left-hand and right-hand limits of the function exist at the point but have different values
• Example of a removable discontinuity: $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$. The limit exists ($\lim_{x \to 1} f(x) = 2$), but the function is undefined at $x = 1$
• Example of a jump discontinuity: $f(x) = \begin{cases} -1 & \text{for } x < 0 \\ 1 & \text{for } x \geq 0 \end{cases}$ at $x = 0$. The left-hand limit ($\lim_{x \to 0^-} f(x) = -1$) and right-hand limit ($\lim_{x \to 0^+} f(x) = 1$) exist but have different values

Infinite Discontinuity

• An infinite discontinuity occurs when the limit of the function approaches positive or negative infinity as $x$ approaches a certain point from either the left or right side
• Infinite discontinuities are associated with vertical asymptotes in the function's graph
• To determine the behavior of the function near an infinite discontinuity, evaluate the one-sided limits of the function as $x$ approaches the point from the left and right
• Example: $f(x) = \frac{1}{x^2}$ has an infinite discontinuity at $x = 0$. As $x$ approaches 0 from either the left or right, the function values approach positive infinity ($\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = +\infty$)