Problem 43 - Calculus

Question

give an example of: A function that is continuous on [0,1] but not continuous on [1,3]

Step-by-Step Solution

Answer

\( f(x) = x \) for [0,1] and \( f(x) = \frac{1}{x-2} \) for [1,3].

Step 1: Understanding the Problem

We need a function that behaves differently in two intervals: continuous on [0,1] and not continuous on [1,3]. Continuous means there are no breaks, jumps, or points where the function goes to infinity within the interval.

Step 2: Define the Function for Interval [0,1]

We choose the function \( f(x) = x \) for the interval [0,1]. This function is continuous throughout this domain since it is a simple linear function without any discontinuities.

Step 3: Modify the Function for Interval [1,3]

To introduce a discontinuity, we modify the function for the interval [1,3] by defining it as follows: \( f(x) = \begin{cases} x, & x \in [0, 1) \ \frac{1}{x-2}, & x \in [1, 3] \end{cases} \). The function \( \frac{1}{x-2} \) becomes undefined and approaches infinity as \( x \) approaches 2, thus making \( f \) discontinuous at \( x = 2 \).

Step 4: Check Continuity on [0,1]

Verify that \( f(x) = x \) is continuous on \([0,1]\). A linear function is continuous everywhere and, particularly, it is continuous over the closed interval [0,1].

Step 5: Check Discontinuity on [1,3]

Check if the function is discontinuous on [1,3]. At \( x = 2 \), \( f(x) = \frac{1}{x-2} \) causes a vertical asymptote, indicating a discontinuity. Thus, the function is not continuous in the interval [1,3].

Key Concepts

Continuous Function

A continuous function is one that can be drawn on a graph without lifting the pencil from the paper. If a function is continuous over a specific interval, it means there are no sudden jumps, breaks, or undefined points within that range.
When we say that a function is continuous on [0,1], it implies that each point in this interval connects smoothly to the next.
Furthermore, both the left and right-hand limits and the actual value at any given point must coincide for it to be continuous. To illustrate, consider the function defined as \( f(x) = x \). This function is a straight line passing through the origin with a consistent slope throughout.
For the interval [0,1], it satisfies all conditions for continuity, meaning it doesn’t disrupt or pause. Understanding continuity is crucial for identifying how functions behave over specified ranges.

Interval Notation

Interval notation is a method used to denote the set of all numbers between two endpoints. In mathematical terms, it signifies the start and stop points of an interval, making it efficient to describe segments on a number line.
Brackets and parentheses are used to convey whether endpoints are included or excluded:

• Square brackets \([a,b]\) denote a closed interval, including both endpoints.
• Parentheses \((a,b)\) signify an open interval, excluding both endpoints.
• A combination of a bracket and a parenthesis, such as \([a,b)\), means the endpoint enclosed by the bracket is included, whereas the one with the parenthesis is not.

For example, the interval [0,1] includes both 0 and 1, forming a closed interval. This notation becomes essential in defining domains where a function is continuous or discontinuous.

Discontinuity

A discontinuity in a function is a type of interruption in a graph where the function's value suddenly changes or becomes undefined. It is essentially a point where a continuous path is broken.
Discontinuities can be identified by looking for breaks, jumps, or vertical asymptotes in a graph.Different types of discontinuity include:

• Jump Discontinuity: The function jumps from one value to another abruptly.
• Infinite Discontinuity: The function approaches infinity at a certain point, often displayed as a vertical asymptote.
• Removable Discontinuity: The limit exists, but the function is not defined at that point.

In our example, the function \( f(x) = \frac{1}{x-2} \) becomes discontinuous at \( x = 2 \) because it results in a vertical asymptote. Thus, the function is not smooth or connected across the entire interval.

Piecewise Function

A piecewise function is defined by multiple sub-functions, each applying to a certain interval of the domain.
This kind of function allows different rules and expressions to be used for separate intervals, providing flexibility in modeling real-world situations and specific conditions.The notation often used for a piecewise function might look like this:\[f(x) = \begin{cases} expr_1, & \text{if } x \text{ is in interval 1} \expr_2, & \text{if } x \text{ is in interval 2} \\end{cases}\]In the context of our problem, the function is defined as \( f(x) = \begin{cases} x, & x \in [0, 1) \ \frac{1}{x-2}, & x \in [1, 3] \end{cases} \).This formulation efficiently describes the continuous behavior in the interval [0,1) and introduces a discontinuity in the interval [1,3]. Understanding how to work with piecewise functions is pivotal for analyzing functions that have different behaviors over different segments of their domain.