https://file.aiquickdraw.com/mathbot/file/17236755257907tlog83v.jpg

Let's analyze the function \( g(x) \) given in the image:

\[
g(x) = \begin{cases} 
x - 2 & \text{if } x \geq 2 \\
2 - x & \text{if } x < 2 
\end{cases}
\]

### Continuity Analysis:

1. **For \( x > 2 \) and \( x < 2 \):**  
   Each piece of the function \( g(x) \) is a linear function, and linear functions are continuous everywhere. Therefore, the function is continuous for all \( x > 2 \) and \( x < 2 \).

2. **At \( x = 2 \):**  
   To determine if \( g(x) \) is continuous at \( x = 2 \), we need to check if:
   \[
   \lim_{x \to 2^-} g(x) = \lim_{x \to 2^+} g(x) = g(2)
   \]

   - The left-hand limit as \( x \) approaches 2 from the left (i.e., \( x \to 2^- \)):
     \[
     \lim_{x \to 2^-} g(x) = \lim_{x \to 2^-} (2 - x) = 2 - 2 = 0
     \]
   - The right-hand limit as \( x \) approaches 2 from the right (i.e., \( x \to 2^+ \)):
     \[
     \lim_{x \to 2^+} g(x) = \lim_{x \to 2^+} (x - 2) = 2 - 2 = 0
     \]
   - The value of \( g(2) \) from the definition:
     \[
     g(2) = 2 - 2 = 0
     \]

   Since the left-hand limit, right-hand limit, and the value at \( x = 2 \) all equal 0, the function \( g(x) \) is continuous at \( x = 2 \).

### Conclusion:
The function \( g(x) \) is continuous **everywhere on the real line**.

Would you like to explore any details further or ask any questions?

Here are some related questions for further study:

1. Why are linear functions always continuous?
2. How can the concept of limits help determine the continuity of piecewise functions?
3. What happens if one of the pieces in a piecewise function is not continuous?
4. What is the importance of checking both the left-hand and right-hand limits for continuity at a specific point?
5. How does this function compare to other piecewise functions in terms of continuity?
6. What does it mean for a function to be continuous on an interval?
7. Can a function be continuous at some points but not at others? Provide an example.
8. How would you graph the function \( g(x) \) to visually confirm its continuity?

**Tip:** When analyzing the continuity of a piecewise function, it's crucial to examine the function at the boundary points where the definition changes.

Explore the continuity analysis of the piecewise function g(x) = { x - 2 if x ≥ 2, 2 - x if x < 2 }. Understand why each piece of this function is continuous and how to determine continuity at specific points. Suitable for advanced high school students and anyone interested in calculus and mathematical analysis.

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