Problem 100 - Single Variable Calculus

Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give \(a n\) example that shows it is false. If \(f\) is continuous on the interval \((1,5)\), then \(f\) is continuous on the interval \((2,4)\).

Step-by-Step Solution

Answer

The statement is true. If a function \(f\) is continuous on the interval (1,5), it must also be continuous on the interval (2,4) because (2,4) is a smaller interval within the larger interval (1,5), and the function \(f\) is continuous at every point in the larger interval, including the points in the smaller interval (2,4).

Step 1: Recall the Definition of Continuity

A function \(f(x)\) is said to be continuous at a point \(x=a\) if the following three conditions are met: 1. \(f(a)\) is defined 2. \(\lim\limits_{x\to a} f(x)\) exists 3. \(\lim\limits_{x\to a} f(x) = f(a)\) A function is continuous on an interval if it is continuous at every point in that interval. #Step 2: Determine if the Statement is True or False#

Step 2: Determine if the Statement is True or False

Since the function f is continuous on the interval (1,5), it means that the function is continuous at every point in this interval. Since the interval (2,4) is a subset of the interval (1,5), this means the function f must also be continuous at every point in the interval (2,4). #Step 3: Confirm and Explain the Result#

Step 3: Confirm and Explain the Result

Based on the previous step, we can conclude that the statement is true. If a function f is continuous on the interval (1,5), then it must be continuous on the interval (2,4) as well because the interval (2,4) is a smaller interval within the larger interval (1,5), and the function f has been proven to be continuous at every point in the larger interval, which includes the points in the smaller interval (2,4).

Key Concepts

Interval Notation

Interval notation is a useful tool in mathematics that helps describe a range of numbers, typically representing intervals on the number line. In interval notation, we use brackets and parentheses to show whether endpoints are included or not included in the interval.

For example, the interval \((1, 5)\) includes all numbers between 1 and 5, but not 1 and 5 themselves. Here, parentheses indicate that the endpoints are not part of the interval (open interval).

If the interval were written as \([1, 5]\), it would include both 1 and 5, indicating a closed interval. Sometimes you might see \([1, 5)\) or \((1, 5]\), which means one endpoint is included while the other is not (half-open).

• \((1, 5)\) - does not include 1 and 5 (open)
• \([1, 5]\) - includes 1 and 5 (closed)
• \([1, 5)\) - includes 1, but not 5
• \((1, 5]\) - includes 5, but not 1

This notation clearly communicates which values are included and is vital for defining the domain of functions or explaining continuity over certain intervals.

Subset

In mathematics, a subset is a set that is entirely contained within another set. To understand this better, let’s imagine we have a set containing all the numbers between 1 and 5, named set A, written as \((1, 5)\). If we have another set B containing all the numbers between 2 and 4, written as \((2, 4)\), we can say that B is a subset of A.

This is because every element in set B also exists in set A. It’s like having a smaller box inside a bigger box. The concept of a subset helps us make broader conclusions about continuity or other properties, as any property verified for set A will hold for B as well.

• Set B: \((2, 4)\)
• Set A: \((1, 5)\)
• Conclusion: B \((2, 4)\) is a subset of A \((1, 5)\)

Understanding subsets is crucial for confirming the properties of functions, especially when discussing continuity.

Limit

The concept of a limit is central in calculus and is crucial for understanding how functions behave. A limit describes the value that a function approaches as the input approaches some point. Imagine being able to "zoom in" on a function graph near a particular point to see where it heads.

For a formal definition, say we have a function \(f(x)\) and we want to find the limit as \(x\) approaches a value \(a\). We express this as \(\lim\limits_{x \to a} f(x)\). If this limit results in a specific number, it means the function values get very close to this number as \(x\) gets closer to \(a\).

• The function might not be defined exactly at \(a\), but that doesn't affect the limit.
• Limits help in defining and understanding function behavior around points not necessarily included in an interval.
• The existence of the limit at \(a\) is one of the conditions for function continuity at that point.

Understanding limits gives insights into how functions progress and helps validate conditions for continuity.

Function Continuity Conditions

A function is continuous at a point if three key conditions are met. These conditions ensure that there are no jumps, breaks, or gaps in the graph of the function at that particular point.

Here's how we decide if a function \(f(x)\) is continuous at a point \(x = a\):

• First, \(f(a)\) must be defined, meaning there should be a value for the function at \(a\).
• Second, the limit \(\lim\limits_{x \to a} f(x)\) must exist. This implies that as \(x\) approaches \(a\) from either direction, the function is approaching a single value.
• Finally, the value obtained from the limit must equal \(f(a)\), denoting that the point on the graph connects smoothly with the neighboring points.

When a function is continuous throughout an interval, it means these conditions hold true at every point within that interval. Being aware of these conditions helps you check for continuity systematically and understand how changes in one part of the interval affect the rest of the function.