`f(x) = {{:(-1, x lt -1),(-x, -1 le x le 1),(1, x gt 1):}` is continous

Options

A. at x=1 but not at x=-1
B. at x=-1 but not at x=1
C. at both x=1 and x=-1
D. at none of x=1 and -1

Answer

To determine the continuity of the piecewise function

\[ f(x) = \begin{cases} -1 & \text{if } x < -1 \\ -x & \text{if } -1 \leq x \leq 1 \\ 1 & \text{if } x > 1 \end{cases} \]

we need to check the continuity at the points where the definition of the function changes, which are \(x = -1\) and \(x = 1\).

### Step 1: Check continuity at \(x = 1\)

To check if \(f(x)\) is continuous at \(x = 1\), we need to verify the following condition:

\[ \lim_{x \to 1^-} f(x) = f(1) = \lim_{x \to 1^+} f(x) \]

**Calculate \(f(1)\):**

Since \(1\) is in the interval \([-1, 1]\), we use the second piece of the function:

\[ f(1) = -1 \]

**Calculate \(\lim_{x \to 1^-} f(x)\):**

As \(x\) approaches \(1\) from the left, we still use the second piece of the function:

\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (-x) = -1 \]

**Calculate \(\lim_{x \to 1^+} f(x)\):**

As \(x\) approaches \(1\) from the right, we use the third piece of the function:

\[ \lim_{x \to 1^+} f(x) = 1 \]

**Conclusion at \(x = 1\):**

We have:

\[ \lim_{x \to 1^-} f(x) = -1, \quad f(1) = -1, \quad \lim_{x \to 1^+} f(x) = 1 \]

Since \(\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)\), \(f(x)\) is not continuous at \(x = 1\).

### Step 2: Check continuity at \(x = -1\)

Now we check for continuity at \(x = -1\):

\[ \lim_{x \to -1^-} f(x) = f(-1) = \lim_{x \to -1^+} f(x) \]

**Calculate \(f(-1)\):**

Since \(-1\) is included in the interval \([-1, 1]\):

\[ f(-1) = -(-1) = 1 \]

**Calculate \(\lim_{x \to -1^-} f(x)\):**

As \(x\) approaches \(-1\) from the left, we use the first piece of the function:

\[ \lim_{x \to -1^-} f(x) = -1 \]

**Calculate \(\lim_{x \to -1^+} f(x)\):**

As \(x\) approaches \(-1\) from the right, we use the second piece of the function:

\[ \lim_{x \to -1^+} f(x) = -(-1) = 1 \]

**Conclusion at \(x = -1\):**

We have:

\[ \lim_{x \to -1^-} f(x) = -1, \quad f(-1) = 1, \quad \lim_{x \to -1^+} f(x) = 1 \]

Since \(\lim_{x \to -1^-} f(x) \neq f(-1)\), \(f(x)\) is not continuous at \(x = -1\).

### Final Conclusion

The function \(f(x)\) is not continuous at both \(x = 1\) and \(x = -1\).