Question

Negate the following sentences. For every positive number \(\varepsilon\), there is a positive number \(\delta\) such that \(|x-a|<\delta\) implies \(|f(x)-f(a)|<\varepsilon\).

Short answer

The negated sentence is: 'There exists a positive number \(\varepsilon\) such that for all positive numbers \(\delta\) either \( |x-a| \geq \delta \), or if \( |x-a|<\delta \) is true then \( |f(x)-f(a)| \geq \varepsilon \)'.

Step-by-step solution

Negate the first part

Start by negating the first part: 'For every positive number \(\varepsilon\)' becomes 'There exists a positive number \(\varepsilon\) such that...'.

Negate the second part

Next, negate the second part: 'there is a positive number \(\delta\) such that...' becomes 'For all positive numbers \(\delta\) it happens that...'.

Negate the logical implication

'\( |x-a|<\delta \) implies \( |f(x)-f(a)|<\varepsilon \)' becomes 'Either \( |x-a|<\delta \) does not hold, that is \( |x-a| \geq \delta \), or \( |x-a|<\delta \) holds but the implication does not hold, that is \( |f(x)-f(a)|\geq\varepsilon \)'.

Key concepts

Logical Implication

Understanding the concept of logical implication is fundamental in mathematical reasoning and proofs. It describes a relationship between two statements, where the truth of the first statement (the premise) guarantees the truth of the second statement (the consequence). This is commonly expressed in the form 'if P, then Q', where P is the premise and Q is the consequence.

In the exercise given, the original statement implied that for any positive number \(\varepsilon\), a \(\delta\) could be found making the second condition true. To negate this, we don't need to prove that the conclusion is always false, but rather that there is at least one instance where the implication does not hold. By finding a single \(\varepsilon\) for which no \(\delta\) satisfies the condition, the entire implication is negated.

Negation can flip the assumed relationship, making it critical in constructing solid arguments or disproving conjectures within mathematical proofs.

Mathematical Proofs

The practice of performing mathematical proofs is at the core of understanding and forwarding mathematical thought. A proof is a logical argument that demonstrates the truth of a given proposition. It includes a series of statements, each following logically from the preceding one leading to the conclusion that the proposition is true.

In our exercise, the process of negation is part of constructing a type of proof known as a proof by contradiction. This method involves assuming the opposite of what we want to prove and then showing that this assumption leads to a contradiction, thereby proving that our original statement must be true. Understanding the structure and logic behind proofs, especially the nuances of logical implication, is essential in mastering advanced mathematics. It is these rigorous methods that secure the foundation upon which all mathematical theory is built.

Epsilon-Delta Definition

The epsilon-delta definition of a limit is a staple in calculus, providing a formal definition of what it means for a function to approach a limit at a particular point. Specifically, it defines the limit of a function \(f(x)\) as \(x\) approaches a value \(a\) in terms of two values: \(\varepsilon\) (epsilon) and \(\delta\) (delta). The definition states that for every \(\varepsilon > 0\), there must exist a \(\delta > 0\) such that if \(0 < |x-a| < \delta\), then \(|f(x)-f(a)| < \varepsilon\).

The epsilon-delta definition requires an arbitrarily small gap around the limit that can be controlled by choosing a corresponding \(\delta\) for every \(\varepsilon\). In the negation of this definition, as per the exercise, we are stating that there exists an \(\varepsilon\) for which no matter how we choose \(\delta\), the function \(f(x)\) will not stay within the \(\varepsilon\)-gap around \(f(a)\) once \(x\) is within the \(\delta\)-neighborhood of \(a\). This concept is essential for understanding the precision required in calculus and solidifying the logic behind limits, continuity, and differentiability.