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find any points discontinuity of these functions

To determine points of discontinuity in this piecewise function, \( f(x) \), we need to check the behavior of \( f(x) \) at the points where the definition of the function changes. Specifically, we look at \( x = -2 \) and \( x = 0 \), as these are the boundaries between the different expressions for \( f(x) \).

The function \( f(x) \) is defined as follows:

\[
f(x) = \begin{cases} 
2 - x & \text{for } x < -2 \\ 
x^2 & \text{for } -2 \leq x \leq 0 \\ 
e^x & \text{for } x > 0 
\end{cases}
\]

### Steps for Checking Discontinuity

1. **Evaluate Left-Hand and Right-Hand Limits at \( x = -2 \):**
   - **Left-hand limit** as \( x \to -2^- \): Use \( f(x) = 2 - x \).
     \[
     \lim_{x \to -2^-} f(x) = 2 - (-2) = 4
     \]
   - **Right-hand limit** as \( x \to -2^+ \): Use \( f(x) = x^2 \).
     \[
     \lim_{x \to -2^+} f(x) = (-2)^2 = 4
     \]
   - **Value of \( f(x) \) at \( x = -2 \)**: According to the definition, \( f(-2) = (-2)^2 = 4 \).

   Since the left-hand limit, right-hand limit, and \( f(-2) \) are all equal to 4, there is no discontinuity at \( x = -2 \).

2. **Evaluate Left-Hand and Right-Hand Limits at \( x = 0 \):**
   - **Left-hand limit** as \( x \to 0^- \): Use \( f(x) = x^2 \).
     \[
     \lim_{x \to 0^-} f(x) = 0^2 = 0
     \]
   - **Right-hand limit** as \( x \to 0^+ \): Use \( f(x) = e^x \).
     \[
     \lim_{x \to 0^+} f(x) = e^0 = 1
     \]
   - **Value of \( f(x) \) at \( x = 0 \)**: According to the definition, \( f(0) = 0^2 = 0 \).

   Here, the left-hand limit as \( x \to 0^- \) is 0, while the right-hand limit as \( x \to 0^+ \) is 1. Since the limits from the left and right do not match, there is a discontinuity at \( x = 0 \).

### Conclusion
The function \( f(x) \) has a point of discontinuity at \( x = 0 \).

Would you like details on how to classify the type of discontinuity, or do you have any other questions?

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#### Related Questions
1. How do we classify different types of discontinuities?
2. What would be the graphical representation of this function?
3. How can we define continuity for functions with multiple intervals?
4. What role do limits play in identifying discontinuities?
5. What are examples of functions with essential (infinite) discontinuities?

#### Tip
When analyzing piecewise functions, always check for continuity at boundary points where the function's expression changes.

Find any points of discontinuity of these functions

Explore how to determine points of discontinuity in a piecewise-defined function. This solution uses limits from the left and right to check continuity at boundary points, providing a comprehensive analysis of where discontinuities occur.

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