Question

Sketch the graph of a function that has domain \([0,2]\) and is continuous on \([0,2)\) but not on \([0,2]\).

Short answer

A continuous line from 0 to just before 2 and a jump to 3 at 2.

Step-by-step solution

Understand the Domain

The function should be defined from 0 to 2 inclusively, i.e., it includes 0 and 2, making it a closed interval at 0. This means it has some sort of behaviour at both ends of the interval.

Ensure Continuity on [0,2)

Make the function continuous from 0 up to 2, not including 2. A simple function like a linear function \( f(x) = x \) on \([0, 2)\) would be continuous because it is a straight line without breaks on this interval.

Introduce Discontinuity at 2

To make the function not continuous at \(x = 2\), introduce a jump or a hole at \(x = 2\). For example, define \( f(2) = 3 \). The function remains \( f(x) = x \) for \( 0 \leq x < 2 \), but has a sudden jump at \(x = 2\).

Combine the Observations

The function is \( f(x) = x \) on \([0, 2)\) and \( f(x) = 3 \) at \(x = 2\). This makes the graph a straight line from 0 to just before 2, and then a point at (2, 3), demonstrating the discontinuity.

Sketch the Graph

Draw the line starting from (0,0) to just before (2,2), indicating continuity up to this point. Then place an open dot at (2,2) and a closed dot at (2,3) to show the discontinuity and the actual value of the function at \(x = 2\).

Key concepts

Understanding Intervals in Mathematics

An interval in mathematics is essentially a range of numbers between two endpoints. There are different types of intervals, each defining how the endpoints are treated. For example, a closed interval \([a,b]\) includes both endpoints, whereas an open interval \( (a,b) \) excludes them. In the given exercise, the domain is \( [0,2] \), meaning the function is defined for all x between 0 and 2, including both of these endpoints.
The way intervals are defined is crucial because it determines how the function behaves at its boundaries. In real-world scenarios, this impacts how continuous or discontinuous a graph may appear at these critical points. In this case, the interval \( [0, 2) \) signifies that the function behaves continuously from 0 up to, but not including, 2. Utilizing intervals effectively makes it easier to predict and visualize the function's behavior across its domain. This deep understanding is vital when sketching and analyzing functions.

Exploring Discontinuities in Functions

Discontinuity refers to a point at which a function does not behave in a smooth or predictable manner. This can be visualized as a 'break' or 'jump' in the graph of the function. There are several types of discontinuities, including jump discontinuity, removable discontinuity, and infinite discontinuity.

In the exercise, a jump discontinuity is introduced at \( x = 2 \). This means that just before reaching this point, the function follows one rule, but exactly at this point, the defined function value differs.
For example, the function remains as \( f(x) = x \) up to but not including 2, creating a straight line. However, at \( x = 2 \), the function value is set to \( f(2) = 3 \), causing the jump. Visually, this can be represented by an open dot at the point \( (2, 2) \) and a filled dot at \( (2, 3) \) on the graph.
Recognizing discontinuities is important because it alters the understanding of the function and affects application in models that demand precision. Properly identifying and representing discontinuities can aid in predicting and managing irregular behaviors in functions.

The Art of Graph Sketching

Graph sketching is an essential skill in mathematics that allows students and professionals to visualize functions and their behaviors. To sketch a graph, especially when discontinuities are involved, one must carefully represent these characteristics.

In this exercise, the task was to sketch a graph on the domain \( [0, 2] \) such that it was continuous on \( [0, 2) \) but not on the entire interval. Here’s a step-by-step visualization strategy:

• Start by drawing a coordinate plane.
• Sketch the line segment of the function \( f(x) = x \) from \( (0, 0) \) to just before \( (2, 2) \), maintaining continuity.
• Place an open dot at \( (2, 2) \) to indicate a hole in continuity at this point.
• Mark a closed dot at \( (2, 3) \) to represent the jump to the actual functional value at \( x = 2 \).

Effective graph sketching involves translating these mathematical instructions into a visual format that captures both continuity and discontinuity, ensuring the graph is both informative and accurate. This visual skill is invaluable in many areas of analysis, problem-solving, and communicating complex mathematical concepts.