Graph \( f \), locate all points of discontinuity, and discuss the behavior of \( f \) at these points. \( f(x)=\left\{\begin{array}{c}-1+x \text { if } x \leq 1 \\ 1-x \text { if } x>1\end{array}\right. \) Find \( f(x) \) at any points of discontinuity. Select the correct choice below and, if necessary, A. The point(s) of discontinuity defined by \( f(x) \) have the value(s) (Use a comma to separate answers as needed.) B. The point(s) of discontinuity are not defined by \( f(x) \). C. There are no points of discontinuity. C.

Let's analyze the function \( f(x) \): \[ f(x) = \begin{cases} -1 + x & \text{if } x \leq 1 \\ 1 - x & \text{if } x > 1 \end{cases} \] ### Step 1: Identifying Potential Points of Discontinuity The function \( f(x) \) is defined piecewise with different expressions for \( x \leq 1 \) and \( x > 1 \). The primary candidate for a point of discontinuity is at \( x = 1 \), where the definition of the function changes. ### Step 2: Checking Continuity at \( x = 1 \) To determine if \( f(x) \) is continuous at \( x = 1 \), we need to evaluate: 1. **Left-hand limit** as \( x \) approaches 1: \[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (-1 + x) = -1 + 1 = 0 \] 2. **Right-hand limit** as \( x \) approaches 1: \[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (1 - x) = 1 - 1 = 0 \] 3. **Value of the function at \( x = 1 \)**: \[ f(1) = -1 + 1 = 0 \] Since both the left-hand limit and the right-hand limit equal the function's value at \( x = 1 \), the function is **continuous** at this point. ### Step 3: Conclusion There are no other points where the definition of \( f(x) \) changes, and within each interval, the function is linear (hence continuous). Therefore, there are **no points of discontinuity** in the function \( f(x) \). **Answer:** C. There are no points of discontinuity.