To find the discontinuities of the piecewise function \( f(x) \) defined as:

\[
f(x) = 
\begin{cases} 
x - 1 & \text{if } x \leq -1, \\
x^2 & \text{if } -1 < x \leq 0, \\
x + 1 & \text{if } x > 0,
\end{cases}
\]

we need to check the continuity of the function at the points where the definition changes, specifically at \( x = -1 \) and \( x = 0 \).

### 1. Check Continuity at \( x = -1 \):

For \( f(x) \) to be continuous at \( x = -1 \), the following must be true:

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

**Left-hand limit as \( x \) approaches \( -1 \) from the left:**

\[
\lim_{x \to -1^-} f(x) = \lim_{x \leq -1, x \to -1^-} (x - 1) = (-1 - 1) = -2
\]

**Right-hand limit as \( x \) approaches \( -1 \) from the right:**

\[
\lim_{x \to -1^+} f(x) = \lim_{-1 < x \leq 0, x \to -1^+} (x^2) = (-1)^2 = 1
\]

**Function value at \( x = -1 \):**

\[
f(-1) = (-1) - 1 = -2
\]

Since \( \lim_{x \to -1^-} f(x) = -2 \) and \( \lim_{x \to -1^+} f(x) = 1 \), the left-hand and right-hand limits are not equal. Therefore, \( f(x) \) is **discontinuous** at \( x = -1 \).

### 2. Check Continuity at \( x = 0 \):

For \( f(x) \) to be continuous at \( x = 0 \), the following must be true:

\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)
\]

**Left-hand limit as \( x \) approaches \( 0 \) from the left:**

\[
\lim_{x \to 0^-} f(x) = \lim_{-1 < x \leq 0, x \to 0^-} (x^2) = (0)^2 = 0
\]

**Right-hand limit as \( x \) approaches \( 0 \) from the right:**

\[
\lim_{x \to 0^+} f(x) = \lim_{x > 0, x \to 0^+} (x + 1) = 0 + 1 = 1
\]

**Function value at \( x = 0 \):**

\[
f(0) = (0)^2 = 0
\]

Since \( \lim_{x \to 0^-} f(x) = 0 \) and \( \lim_{x \to 0^+} f(x) = 1 \), the left-hand and right-hand limits are not equal. Therefore, \( f(x) \) is **discontinuous** at \( x = 0 \).

### Conclusion

The function \( f(x) \) is discontinuous at:
- \( x = -1 \)
- \( x = 0 \)

These are the points where the function changes its form and fails to satisfy the continuity conditions.

Would you like any further details or clarifications?

Here are 5 questions related to this topic:

1. How can we determine whether a function is continuous at a given point?
2. What are the types of discontinuities a function can have?
3. How do we find the derivative of a piecewise function?
4. What is the difference between a removable discontinuity and a jump discontinuity?
5. Can a function be continuous but not differentiable at certain points?

**Tip:** Always check both the left-hand and right-hand limits when analyzing the continuity of a piecewise function at the boundary points where the function's definition changes.

Discover how to find discontinuities in a piecewise function analysis. This detailed explanation covers the conditions for continuity at critical points and provides step-by-step examples. Suitable for high school students studying functions and limits.

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