https://file.aiquickdraw.com/mathbot/file/1722454881492psq0slaa.jpeg

Here is the graph of the function \( f(x) \) defined by:

\[
f(x) = \begin{cases} 
1 - x^2, & \text{if } x \neq 1 \\
2, & \text{if } x = 1 
\end{cases}
\]

- The green curve represents \( 1 - x^2 \) for \( x \neq 1 \), which is a downward-opening parabola.
- The red dot at \( (1, 2) \) indicates the value of \( f(x) \) at \( x = 1 \), showing the discontinuity.

### Summary:
- **Left-hand limit**: \( \lim_{x \to 1^-} f(x) = 0 \)
- **Right-hand limit**: \( \lim_{x \to 1^+} f(x) = 0 \)
- **Two-sided limit**: \( \lim_{x \to 1} f(x) = 0 \)

The function \( f(x) \) is discontinuous at \( x = 1 \) because \( f(1) = 2 \), which is different from the limit value (0).

If you have any questions or need further explanations, feel free to ask!

### Further Questions:
1. How can you determine if a function is continuous at a point?
2. What is the significance of left-hand and right-hand limits in calculus?
3. Why is it important to consider the value of the function at a point when discussing continuity?
4. How would the graph of \( f(x) \) change if \( f(1) = 0 \) instead?
5. Can you give an example of a real-world scenario where a piecewise function might be used?
6. What is the derivative of the function \( 1 - x^2 \)?
7. How does the concept of limits help in understanding instantaneous rates of change?
8. What is the difference between a removable discontinuity and a jump discontinuity?

### Tip:
To determine the continuity of a function at a point, ensure that the left-hand limit, right-hand limit, and the value of the function at that point are all equal.

Explore the concept of discontinuity in piecewise functions through an example involving the function defined as f(x) = 1 - x^2 for x ≠ 1 and f(x) = 2 for x = 1. Learn about left-hand, right-hand, and two-sided limits. Suitable for advanced high school students.

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