Problem 50 - Calculus: Early Transcendentals

Question

Suppose that a function \( f \) is continuous on \( [0, 1] \) except at 0.25 and that \( f(0) = 1 \) and \( f(1) = 3 \). Let \( N = 2 \). Sketch two possible graphs of \( f \), one showing that \( f \) might not satisfy the conclusion of the Intermediate Value Theorem and one showing that \( f \) might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn't satisfy the hypothesis.)

Step-by-Step Solution

Answer

One function can jump at 0.25, missing 2, and the other can pass through 2 despite the jump.

Step 1: Understand the Problem

The task involves sketching two graphs of a function \( f \) that is continuous on the interval \([0, 1]\) except at the point \( x = 0.25 \), with given function values \( f(0) = 1 \) and \( f(1) = 3 \). We need to show one graph where the function does not meet the Intermediate Value Theorem (IVT) and one where it still does, despite not satisfying the continuity hypothesis of the theorem.

Step 2: Review the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function \( f \) is continuous on \([a, b]\) and \( N \) is any number between \( f(a) \) and \( f(b) \), then there is at least one number \( c \) in \((a, b)\) such that \( f(c) = N \). For \([0, 1]\) with \( f(0) = 1 \) and \( f(1) = 3 \), the IVT suggests that \( f \) should take on every value between 1 and 3.

Step 3: Sketch a Graph Not Satisfying IVT

To show \( f \) not satisfying the IVT, sketch a graph of \( f \) where it jumps at \( x = 0.25 \). An example is if \( f(x) = 2 \) for \( x < 0.25 \) and \( f(x) = 3 \) for \( x \geq 0.25 \). This function does not achieve the value 2 on \((0, 1)\), hence it doesn't satisfy the IVT for \( N = 2 \).

Step 4: Sketch a Graph Still Satisfying IVT

To show \( f \) still satisfying the IVT, sketch the graph where \( f \) is discontinuous at \( x = 0.25 \), but still passes through 2. For example, set \( f(x) = 1 + 2x \) for \( x < 0.25 \) and \( f(x) = 1 + 2(x - 0.5) \) for \( x \geq 0.25 \). This causes \( f(x) \) to jump but still pass through the value 2.

Key Concepts

continuity

Continuity is a fundamental concept in calculus and mathematical analysis. It refers to the behavior of functions where small changes in the input result in small changes in the output. In layman's terms, if you can draw a function without lifting your pencil from the paper, it is continuous on that interval.
This concept helps us understand how functions behave over an interval. If a function is continuous on an interval, it means there are no sudden jumps or gaps, which makes the function predictable and smooth.
Continuous functions are crucial in mathematical modeling and real-life scenarios as they allow us to interpolate and predict values within the domain of the function without encountering unforeseen disruptions.
When you hear that a function is continuous on an interval, it means every point within the interval is connected. There are no breaks in the value of the function.

functions

Functions are at the heart of mathematics, providing a formal way to describe relationships between two sets of data or quantities. A function, often denoted by symbols such as \( f \), relates each element from a set called the domain to a single element in another set called the codomain.
The notation \( f(x) \) represents the output of function \( f \) given an input \( x \). It is a concise way to express complex relationships and can be as simple as \( f(x) = x^2 \) or more intricate, depending on the application.
Functions serve multiple purposes:

• Modeling real-world phenomena, allowing prediction and analysis.
• Providing a foundation for areas like calculus, algebra, and more.
• Describing curves, surfaces, and other geometric objects.

Understanding how functions work is vital for solving mathematical problems, as they provide clear pathways to determining outcomes based on given inputs.

continuous functions

Continuous functions are a specific type of function characterized by their smoothness over an interval. This smoothness implies that there are no breaks, jumps, or holes in the function's graph within a given range. In mathematical terms, a function \( f \) is continuous at a point \( a \) if:

• The function \( f(a) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( a \) is equal to \( f(a) \).

This description ensures that around every point \( a \), the behavior of the function adheres to the expected value. Continuous functions are predictable and thus crucial for analysis, particularly in the context of the Intermediate Value Theorem.
These functions play an essential role in mathematics because they are often the easiest to work with and model. They underpin many theorems and are foundational for understanding calculus and other advanced subjects.

discontinuity

Discontinuity occurs when a function does not meet the smooth and unbroken characteristics of a continuous function. It is a point or set of points where a function "jumps" or has gaps. In our exercise, there is a discontinuity at \( x = 0.25 \), meaning \( f \) behaves unexpectedly at this point, deviating from the continuous path.
There are different types of discontinuities:

• Jump Discontinuity: The function has a visible "jump" from one value to another.
• Infinite Discontinuity: The function heads towards infinity around the point of discontinuity.
• Removable Discontinuity: A point that could be "fixed" by re-defining the function value at that point.

These discontinuities are crucial when considering the Intermediate Value Theorem, as they can prevent the theorem's conditions from being satisfied, making the function not pass through certain values without precise formulation.
Understanding discontinuities allows us to better comprehend the limitations and boundaries of functions, particularly when ensuring predictions and calculations are accurate within specific intervals.