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Let's analyze the function \( f(x) \) and determine whether it is continuous at \( a = 2 \).

The function is given as:

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

To check continuity at \( x = 2 \), we need to verify the following conditions:

1. **Existence of \( f(2) \)**:
   \[
   f(2) = 2(2) + 1 = 5
   \]

2. **Existence of the left-hand limit as \( x \to 2^- \)** (i.e., approaching 2 from the left):
   \[
   \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (1 - x^2) = 1 - 2^2 = 1 - 4 = -3
   \]

3. **Existence of the right-hand limit as \( x \to 2^+ \)** (i.e., approaching 2 from the right):
   \[
   \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (2x + 1) = 2(2) + 1 = 5
   \]

4. **Continuity condition**: A function is continuous at \( x = 2 \) if:
   \[
   \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2)
   \]
   Here, we observe that:
   \[
   \lim_{x \to 2^-} f(x) = -3 \quad \text{and} \quad \lim_{x \to 2^+} f(x) = 5
   \]
   Since \( -3 \neq 5 \), the left-hand and right-hand limits are not equal. Therefore, the function is **not continuous** at \( x = 2 \).

### English Explanation:
The function \( f(x) \) is not continuous at \( x = 2 \) because the left-hand limit as \( x \) approaches 2 is \(-3\), while the right-hand limit is 5. Since these two limits are not equal, and the value of the function at \( x = 2 \) is 5, the function has a discontinuity at this point.

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Would you like more details or have any questions?

Here are 5 related questions for further understanding:
1. What is the definition of a continuous function?
2. How do we compute one-sided limits for piecewise functions?
3. Why does the value of \( f(2) \) not influence the left-hand limit?
4. How can we graphically represent discontinuities in piecewise functions?
5. What types of discontinuities can a function have?

**Tip**: Always verify both one-sided limits when checking for continuity at a point in piecewise functions.

Q5 - Explain why or why not the function is continuous at a = 2. a. Use Math Notation b. Provide an English sentence. f(x) = {1 - x^2 if 0 ≤ x < 2; 2x + 1 if x ≥ 2}

Explore the continuity of the function f(x) at x = 2. Understand the detailed step-by-step analysis using limits, left-hand and right-hand calculations, and continuity criteria. Suitable for Grades 11-12 students studying advanced calculus.

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