https://file.aiquickdraw.com/mathbot/file/1726492895498u7pmfpvb.jpg

The problem in the image asks for two tasks:

1. Write the mathematical definition of continuity at a point \( a \in A \subseteq \mathbb{R} \) using quantifiers.
2. Negate the statement.

### Step 1: Writing the definition of continuity using quantifiers

A function \( f \) is said to be continuous at a point \( a \in A \subseteq \mathbb{R} \) if, for each positive real number \( \epsilon \), there exists a positive real number \( \delta \) such that:

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

This is the formal statement of continuity at the point \( a \), expressed using quantifiers.

### Step 2: Negating the statement

To negate the statement, we need to reverse the quantifiers and modify the conditions accordingly. The negation of the above statement becomes:

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

This is the negation of the continuity definition, which states that there exists some \( \epsilon \) such that no matter how small \( \delta \) is, there is always some \( x \) such that \( |f(x) - f(a)| \) is not smaller than \( \epsilon \). In other words, the function is not continuous at \( a \).

Would you like further clarification or details on this explanation?

### Related Questions:
1. What does it mean for a function to be continuous in terms of limits?
2. How does the concept of continuity extend to higher dimensions (e.g., multivariable calculus)?
3. What are some common examples of discontinuous functions?
4. How does the epsilon-delta definition of continuity relate to intuitive ideas of smoothness or no breaks in a function?
5. Can a function be continuous at a single point but discontinuous elsewhere?

### Tip:
When negating a quantified statement, reverse both the quantifiers and the inequalities/conditions within the statement.

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.

Explore the epsilon-delta definition of continuity for a function at a point. This problem includes writing the definition using quantifiers and negating the statement to demonstrate the function's discontinuity. Suitable for undergraduate students studying real analysis.

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