Question

A function \(y=f(x)\) is discontinuous at \(x=x_{0}\) when any of the three requirements for continuity is violated at \(x=x_{0} .\) Construct three graphs to illustrate the violation of each of those requirements.

Short answer

Three graphs show discontinuity: a hole at \(x=1\), a jump at \(x=x_0\), and a mismatch at \(x=1\).

Step-by-step solution

Understanding Continuity Requirements

A function is continuous at a point \(x = x_0\) when it satisfies three conditions: 1) the function is defined at \(x = x_0\), which means \(f(x_0)\) exists, 2) the limit as \(x\) approaches \(x_0\) exists, and 3) the limit as \(x\) approaches \(x_0\) is equal to \(f(x_0)\). If any of these conditions are violated, the function is discontinuous at \(x_0\).

Graph 1: Violation of the First Requirement

The first requirement is \(f(x_0)\) exists. Consider a function where there is a hole or gap at \(x = x_0\). This could occur if, for instance, \(f(x) = \frac{x^2 - 1}{x - 1}\). At \(x = 1\), the expression is undefined due to division by zero, causing discontinuity. Graph this function and show a hole at \(x = 1\).

Graph 2: Violation of the Second Requirement

The second requirement is that \(\lim_{x \to x_0} f(x)\) exists. This means the limit from the left and right should be equal. A typical case is a step function such as \(f(x) = \begin{cases} 1, & \text{if } x < x_0 \ 2, & \text{if } x \ge x_0 \end{cases}\). At \(x = x_0\), the left-hand limit \(\lim_{x \to x_0^-} f(x) = 1\) and the right-hand limit \(\lim_{x \to x_0^+} f(x) = 2\), so the overall limit \(\lim_{x \to x_0} f(x)\) does not exist, causing discontinuity. Graph this to show a jump at \(x = x_0\).

Graph 3: Violation of the Third Requirement

The third requirement is \(\lim_{x \to x_0} f(x) = f(x_0)\). Consider a function where the limit exists and \(f(x_0)\) exists, but they are unequal. Take \(f(x) = \begin{cases} x^2, & \text{if } x eq 1 \ 3, & \text{if } x = 1 \end{cases}\). At \(x = 1\), \(f(1) = 3\), but \(\lim_{x \to 1} f(x) = 1^2 = 1\), causing a discontinuity. Graph this to indicate a mismatch at \(x = 1\).

Key concepts

Discontinuous Functions

Discontinuous functions are an important aspect of mathematical continuity. A function is considered discontinuous at a specific point if it fails to meet any one of the three continuity requirements at that point. To understand this better, we first need to break down these continuity conditions:

• The function must be defined at that point, meaning there is a specific output corresponding to the input.
• The limit of the function as it approaches that point must exist, so the left-hand and right-hand limits are equal.
• The limit as the function approaches that point should equal the function's value at that point.

When a function violates any of these rules at a given point, it results in a discontinuity. Discontinuities can appear as gaps, jumps, or mismatches in function values and are crucial in identifying the behavior of functions. Graphs can visually represent these discontinuities, helping students identify where and how the function fails to be continuous.

Limit of a Function

The concept of the limit of a function is central to understanding continuity and the behavior of functions as they approach specific points. When considering the limit, we observe how a function behaves as it gets closer to a specific input value. There are two kinds of limits that should match for continuity:

• **Left-hand limit**: The value a function approaches as you come towards the point from the left direction.
• **Right-hand limit**: The value approached from the right direction as you near the point.

For a limit to exist at a point, both the left-hand and right-hand limits must be equal. If there is a discrepancy between these limits, the function is said to be discontinuous at that point. Discontinuous functions can have limits that do not exist due to such mismatches, often resulting in a sudden jumps or steps in a graph. Understanding limits is key to mastering topics in calculus and mathematical analysis.

Graphical Representation of Discontinuity

Visualizing discontinuity through graphs helps to better comprehend where and why a function isn't continuous. Here are three typical graphical scenarios illustrating discontinuity:

• **Hole or Gap**: If a function is missing a value at a point, such as it being undefined, there will appear a hole in the graph. This visually shows the point of discontinuity.
• **Jump Discontinuity**: Often seen in piecewise functions, there may be a sudden jump from one value to another when there is a mismatch between left-hand and right-hand limits. This jump is visibly clear on the graph as a marked vertical leap between two points.
• **Mismatch of Limit and Value**: Even if a limit exists and the function value at the point exists, when they are not equal, a graph will show a point either below or above the function curve, indicating discontinuity.

Graphs make it easier to see the impact of each type of discontinuity, providing a clear picture of where function behavior may change unexpectedly. They are an invaluable tool in analyzing and interpreting mathematical functions.