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To solve for the values of \( x \) for which the function is continuous, let's examine the function provided:

\[
f(x) = \frac{|x - 13|}{x - 13}
\]

### Step 1: Analyze the denominator
The function will be undefined wherever the denominator is zero. So, set the denominator equal to zero:

\[
x - 13 = 0
\]
\[
x = 13
\]

Thus, the function is **undefined at \( x = 13 \)**, and we must exclude \( x = 13 \) from the domain to ensure the function is continuous.

### Step 2: Behavior of the function for \( x \neq 13 \)
We need to check what happens to the function for values of \( x \) on either side of \( x = 13 \):

- **Case 1: \( x > 13 \)**
  
  If \( x > 13 \), then \( |x - 13| = x - 13 \), so the function becomes:

  \[
  f(x) = \frac{x - 13}{x - 13} = 1
  \]

- **Case 2: \( x < 13 \)**

  If \( x < 13 \), then \( |x - 13| = -(x - 13) \), so the function becomes:

  \[
  f(x) = \frac{-(x - 13)}{x - 13} = -1
  \]

Thus, the function is 1 for all \( x > 13 \) and -1 for all \( x < 13 \).

### Step 3: Discontinuity at \( x = 13 \)
At \( x = 13 \), the function has a discontinuity because the function is undefined there and the limits from the left and right do not match:

- As \( x \to 13^+ \), \( f(x) = 1 \)
- As \( x \to 13^- \), \( f(x) = -1 \)

This jump discontinuity at \( x = 13 \) confirms that the function is not continuous at \( x = 13 \).

### Conclusion
The function is continuous for all \( x \neq 13 \). The correct answer is:

\[
\boxed{x \neq 13}
\]

Would you like more details, or do you have any further questions?

### Follow-up Questions:
1. What is the definition of a jump discontinuity in a function?
2. Can the absolute value in the numerator affect the continuity in any other cases?
3. How do you find the limits of a piecewise function like this?
4. Why is the function undefined at \( x = 13 \)?
5. How can we generalize the approach for functions involving absolute values and rational expressions?

### Tip:
When analyzing the continuity of a function with absolute values and rational terms, always check the denominator for zeros and investigate behavior around those points carefully.

Find the values of x for which the function is continuous. f(x) = |x - 13| / (x - 13)

This math problem focuses on determining the continuity of the function f(x) = |x - 13| / (x - 13). The solution explores key mathematical concepts such as continuity, rational functions, and absolute values, and explains why the function is discontinuous at x = 13.

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