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The problem asks to provide an example of a function \( f \) for which the following statement is false:

**Statement:**
If \( |f(x) - l| < \varepsilon \) when \( 0 < |x - a| < \delta \), then \( |f(x) - l| < \varepsilon/2 \) when \( 0 < |x - a| < \delta/2 \).

This statement is essentially related to the formal definition of the limit of a function \( f(x) \) at \( x = a \), but it makes a stronger claim: if the function is close to the limit \( l \) within \( \varepsilon \) over a certain interval \( \delta \), then it should be even closer (within \( \varepsilon/2 \)) over a smaller interval \( \delta/2 \). 

To find an example where this statement is false, we need to find a function where, although it gets within \( \varepsilon \) for the interval \( \delta \), it does not get closer (within \( \varepsilon/2 \)) over the smaller interval \( \delta/2 \).

### Example:

Consider the function:

\[
f(x) =
\begin{cases}
1, & \text{if } x = a, \\
0, & \text{if } x \neq a.
\end{cases}
\]

This function, called the **Dirichlet function** at a single point \( a \), does not satisfy the required condition. Here's why:

1. If \( x \neq a \), then \( f(x) = 0 \), so \( |f(x) - 0| = 0 \), which is smaller than any \( \varepsilon > 0 \). This satisfies the first condition for some \( \delta \).
   
2. However, for smaller intervals (such as \( 0 < |x - a| < \delta/2 \)), the function remains \( 0 \) except exactly at \( x = a \), where it suddenly jumps to 1. Hence, the function does not get closer to 0 within \( \varepsilon/2 \), because at \( x = a \), \( |f(x) - 0| = 1 \), which is not less than \( \varepsilon/2 \).

This shows that the condition \( |f(x) - l| < \varepsilon/2 \) when \( 0 < |x - a| < \delta/2 \) is violated.

Would you like more details or have any questions about this? Here are five related questions you might explore:

1. How does the Dirichlet function behave at discontinuities?
2. Can a continuous function violate the statement, or must it be discontinuous?
3. What does the formal definition of a limit say about the behavior of a function near a point?
4. How does the concept of uniform continuity relate to this problem?
5. What are the implications of the failure of this statement in practical applications of limits?

**Tip:** When working with limits, always consider the behavior of the function on increasingly smaller intervals to fully understand its local behavior.

Give an example of a function f for which the following assertion is false: If |f(x) - l| < ε when 0 < |x - a| < δ, then |f(x) - l| < ε/2 when 0 < |x - a| < δ/2.

Explore a counterexample to the epsilon-delta definition of a limit condition. This example demonstrates how a function may violate the condition where |f(x) - l| < ε/2 does not hold even though |f(x) - l| < ε is satisfied for a smaller interval. Understand this through the Dirichlet function and related concepts in calculus.

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