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The problem in the image is asking you to express the definition of continuity using mathematical quantifiers, and then to negate the statement.

### Part 1: Write the statement using quantifiers

The statement is:  
*A function \( f \) is continuous at a point \( a \in A \subseteq \mathbb{R} \) if for each positive real number \( \epsilon \), we can find a positive real number \( \delta \) such that \( |f(x) - f(a)| < \epsilon \) whenever \( |x - a| < \delta \).*

This can be expressed using quantifiers as:

\[
\forall \epsilon > 0, \exists \delta > 0 \text{ such that } (|x - a| < \delta \implies |f(x) - f(a)| < \epsilon)
\]

### Part 2: Negate the statement

To negate this, you apply negation to each part of the quantifiers and logical statements. The negation of the statement becomes:

\[
\exists \epsilon > 0 \text{ such that } \forall \delta > 0, (|x - a| < \delta \text{ and } |f(x) - f(a)| \geq \epsilon)
\]

This negation means that there exists some \( \epsilon \) such that no matter how small you make \( \delta \), you can always find some \( x \) within the \( \delta \)-neighborhood of \( a \) where the difference \( |f(x) - f(a)| \) is not less than \( \epsilon \). In other words, this expresses discontinuity at \( a \).

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Would you like more details on any part of this explanation, or need help with related topics? Here are five questions to consider next:

1. What is the difference between pointwise and uniform continuity?
2. How does the epsilon-delta definition change for different types of limits?
3. Can you provide an example of a function that is discontinuous and apply this negated definition?
4. How do we define continuity for multivariable functions?
5. How does the concept of continuity relate to differentiability?

**Tip**: When negating mathematical statements, switch the quantifiers (i.e., change "for all" to "there exists" and vice versa), and flip the inequalities or logical statements.

A function f is continuous at a point a ∈ A ⊆ ℝ if for each positive real number ε we can find a positive real number δ such that |f(x) - f(a)| < ε whenever |x - a| < δ. Write this statement using quantifiers and then negate the statement.

This problem explores the epsilon-delta definition of continuity, guiding you through how to express it using mathematical quantifiers and negate the statement to understand discontinuity.

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