Problem 66 - Thomas Calculus

Question

The sign-preserving property of continuous functions Let \(f\) be defined on an interval \((a, b)\) and suppose that \(f(c) \neq 0\) at some \(c\) where \(f\) is continuous. Show that there is an interval \((c-\delta, c+\delta)\) about \(c\) where \(f\) has the same sign as \(f(c)\) .

Step-by-Step Solution

Answer

There exists an interval around \(c\) where \(f(x)\) maintains the sign of \(f(c)\).

Step 1: Understanding the Problem

We want to show that if a continuous function \(f\) does not have a zero at a point \(c\), there exists a small interval around \(c\), in which the function \(f\) does not change its sign. This utilizes the property of continuity to guarantee that \(f\) stays consistently positive or negative around \(c\).

Step 2: Definition of Continuity

Recall that a function \(f\) is continuous at a point \(c\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \( |x - c| < \delta \), it follows that \( |f(x) - f(c)| < \epsilon \).

Step 3: Choosing \(\epsilon\)

Given that \(f(c) eq 0\), choose \(\epsilon = \frac{|f(c)|}{2}\). This choice will help us find the interval \((c-\delta, c+\delta)\) where \(f(x)\) has the same sign as \(f(c)\).

Step 4: Finding \(\delta\)

By the definition of continuity at a point \(c\), with \(\epsilon = \frac{|f(c)|}{2}\), there exists a \(\delta > 0\) such that if \(|x - c| < \delta\), then \(|f(x) - f(c)| < \frac{|f(c)|}{2}\).

Step 5: Simplifying \(|f(x) - f(c)| < \frac{|f(c)|}{2}\)

Since \(|f(x) - f(c)| < \frac{|f(c)|}{2}\), this implies:\[ -\frac{|f(c)|}{2} < f(x) - f(c) < \frac{|f(c)|}{2} \]Adding \(f(c)\) to all sides yields:\[ f(c) - \frac{|f(c)|}{2} < f(x) < f(c) + \frac{|f(c)|}{2} \]

Step 6: Checking the Sign of \(f(x)\)

Observe the interval boundaries:- If \(f(c) > 0\), then \(f(c) - \frac{|f(c)|}{2} > 0\), so \(f(x) > 0\).- If \(f(c) < 0\), then \(f(c) + \frac{|f(c)|}{2} < 0\), so \(f(x) < 0\).Thus, \(f(x)\) maintains the sign of \(f(c)\).

Step 7: Conclusion

Therefore, there exists an interval \((c-\delta, c+\delta)\) in which \(f(x)\) has the same sign as \(f(c)\), either positive if \(f(c) > 0\) or negative if \(f(c) < 0\). This completes our proof using the continuity of the function.

Key Concepts

Sign-Preserving Property

A sign-preserving property in continuous functions is a fascinating concept. If you have a function, say \(f\), that is continuous at a particular point \(c\), and \(f(c)\) is not zero, this property guarantees that in a small neighborhood around \(c\), the function \(f\) keeps the same sign as \(f(c)\). This could mean the function remains positive or negative.
This all relies on the behavior of continuity. Because the function doesn’t have a sharp turn or jump, one can expect that around \(c\), \(f(x)\) will not cross the x-axis, preserving its positivity or negativity. For instance:

• If \(f(c) > 0\), then in a small segment around \(c\), say a distance \(\delta\) away, \(f(x)\) remains greater than zero.
• If \(f(c) < 0\), \(f(x)\) stays less than zero for points near \(c\).

Understanding this property reinforces why continuity (no jumps or breaks) of a function is such a powerful concept in mathematics.

Epsilon-Delta Definition

The epsilon-delta definition is a vital part of understanding continuity mathematically. When we say a function \(f\) is continuous at a point \(c\), intuitively, it means there are no sudden changes in the function's value at \(c\). The epsilon-delta definition formalizes this idea.
Here's how it works:

• Given any \(\epsilon > 0\), there is a \(\delta > 0\) such that whenever \(|x - c| < \delta\), the inequality \(|f(x) - f(c)| < \epsilon\) holds.
• This setup ensures for any small \(\epsilon\), a corresponding \(\delta\) can be found, keeping \(f(x)\) close to \(f(c)\). It effectively controls how much \(f(x)\) can deviate from \(f(c)\).

This method is particularly useful in proofs, as demonstrated in the original problem. By choosing \(\epsilon\) as \(\frac{|f(c)|}{2}\), it ensures the function definitely holds a sign.

Interval Analysis

Interval analysis helps us investigate how functions behave over specific sections of their domain. By analyzing intervals where functions maintain certain properties, we gather more information about continuity and related behavior.
Consider the case where you determine an interval \((c-\delta, c+\delta)\). In this segment:

• You observed the sign of \(f(x)\) will not change, upholding what the sign-preserving property predicts.
• It allows reviewing other functions’ features, such as monotonicity and potential extremities near \(c\).

Through interval analysis, we take a closer look at the function's stability and identify regions where its characteristics unify with theoretical expectations. This makes analysis clearer and computations more directed.