Question

Let \( f \) be the function defined by:
\[ f(x) = \begin{cases} \frac{x^2 - 1}{x^2 - 2|x-1| - 1}, & \text{if } x \neq 1, \\ \frac{1}{2}, & \text{if } x = 1. \end{cases} \] The function is continuous at:

• A. The function is continuous for all values of \( x \)
• B. The function is continuous only for \( x>1 \)
• C. The function is continuous at \( x = 1 \)
• D. The function is not continuous at \( x = 1 \)

Hint:

For a function to be continuous at a point, the left-hand limit, right-hand limit, and the function's value at that point must all exist and be equal.