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The problem asks to explain why the function is discontinuous at the given number \(a = 1\) and sketch the graph of the function. The function is given as:

\[
f(x) = \begin{cases} 
\frac{x^2 - x}{x^2 - 1} & \text{if } x \neq 1 \\
1 & \text{if } x = 1
\end{cases}
\]

### Step-by-Step Solution

#### 1. Discontinuity at \(x = 1\)

To determine whether the function is continuous at \(x = 1\), we need to check the following three conditions:

1. **The function is defined at \(x = 1\):**
   \[
   f(1) = 1
   \]
   So, the function is defined at \(x = 1\).

2. **The limit as \(x\) approaches 1 exists:**
   We need to evaluate the limit as \(x \to 1\) for the expression \( \frac{x^2 - x}{x^2 - 1} \) when \(x \neq 1\). We simplify the expression:

   \[
   \frac{x^2 - x}{x^2 - 1} = \frac{x(x - 1)}{(x - 1)(x + 1)} = \frac{x}{x + 1}, \quad \text{for } x \neq 1
   \]

   Now, we find the limit as \(x\) approaches 1:

   \[
   \lim_{x \to 1} \frac{x}{x + 1} = \frac{1}{1 + 1} = \frac{1}{2}
   \]

   So, the limit as \(x \to 1\) is \( \frac{1}{2} \).

3. **The limit must equal the function's value at \(x = 1\):**
   We have:
   \[
   \lim_{x \to 1} f(x) = \frac{1}{2} \quad \text{and} \quad f(1) = 1
   \]
   Since \( \frac{1}{2} \neq 1 \), the function is **discontinuous** at \(x = 1\).

#### Conclusion
The function is discontinuous at \(x = 1\) because the limit of \(f(x)\) as \(x \to 1\) does not equal the value of the function at \(x = 1\).

### Graph Sketch
The function is piecewise-defined. For \(x \neq 1\), it behaves like \( \frac{x}{x+1} \), which is a rational function. There will be a removable discontinuity at \(x = 1\), where the value jumps from \( \frac{1}{2} \) (the limit) to \(1\) (the actual function value).

Would you like further details or a more in-depth explanation on any step?

### Relative Questions
1. What are the conditions for a function to be continuous at a point?
2. How can you determine the type of discontinuity for a given function?
3. How does simplifying rational expressions help in evaluating limits?
4. What is the graphical significance of removable discontinuities?
5. How can piecewise functions be represented graphically?

### Tip:
Always check for removable discontinuities by factoring and simplifying rational expressions before evaluating limits.

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x) = (x^2 - x) / (x^2 - 1) if x ≠ 1, and f(x) = 1 if x = 1. Determine if the function is discontinuous at a = 1.

Learn why the function f(x) = (x^2 - x) / (x^2 - 1) for x ≠ 1 and f(x) = 1 for x = 1 is discontinuous at a = 1. This problem explores key concepts in limits and continuity, providing a detailed step-by-step solution and graph sketch. Suitable for students in advanced high school or college calculus courses.

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