A function f(x) is defined by, `f(x) = {{:(([x^(2)]-1)/(x^(2)-1)",","for",x^(2) ne 1),(0",","for",x^(2) = 1):}` Discuss the continuity of f(x) at x = 1.

Answer

To determine the continuity of the function \( f(x) \) at \( x = 1 \), we need to check the following conditions:

1. The left-hand limit \( \lim_{x \to 1^-} f(x) \)
2. The right-hand limit \( \lim_{x \to 1^+} f(x) \)
3. The value of the function at that point \( f(1) \)

A function is continuous at a point if the left-hand limit, right-hand limit, and the value of the function at that point are all equal.

### Step 1: Finding the left-hand limit \( \lim_{x \to 1^-} f(x) \)

For \( x < 1 \), we use the definition of \( f(x) \):
\[ f(x) = \frac{x^2 - 1}{x^2 - 1} \]
This simplifies to:
\[ f(x) = 1 \quad \text{for } x^2 \neq 1 \]

Now, we calculate the left-hand limit as \( x \) approaches 1 from the left:
\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} 1 = 1 \]

### Step 2: Finding the right-hand limit \( \lim_{x \to 1^+} f(x) \)

For \( x > 1 \), the function is still defined as:
\[ f(x) = \frac{x^2 - 1}{x^2 - 1} = 1 \quad \text{for } x^2 \neq 1 \]

Now, we calculate the right-hand limit as \( x \) approaches 1 from the right:
\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} 1 = 1 \]

### Step 3: Finding the value of the function at \( x = 1 \)

From the definition of the function, when \( x = 1 \):
\[ f(1) = 0 \]

### Step 4: Conclusion

Now we compare the limits and the value of the function:
- Left-hand limit: \( \lim_{x \to 1^-} f(x) = 1 \)
- Right-hand limit: \( \lim_{x \to 1^+} f(x) = 1 \)
- Value of the function: \( f(1) = 0 \)

Since the left-hand limit and right-hand limit are equal, but both are not equal to \( f(1) \):
\[ \lim_{x \to 1^-} f(x) \neq f(1) \]
Thus, \( f(x) \) is discontinuous at \( x = 1 \).

### Final Answer:
The function \( f(x) \) is discontinuous at \( x = 1 \).

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