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The function \( f(x) \) given in the problem is a piecewise-defined function:

\[
f(x) = \begin{cases} 
x - 4 & \text{if } x < 2 \\ 
-2x + 2 & \text{if } x \geq 2 
\end{cases}
\]

### Analysis:
1. **Graph each piece**:
   - For \( x < 2 \), the function is \( f(x) = x - 4 \), which is a line with slope 1 and y-intercept -4. This segment is defined for \( x < 2 \).
   - For \( x \geq 2 \), the function is \( f(x) = -2x + 2 \), which is a line with slope -2 and y-intercept 2. This segment is defined for \( x \geq 2 \).

2. **Check for continuity at \( x = 2 \)**:
   - **Left-hand limit**: \( \lim_{x \to 2^-} f(x) = 2 - 4 = -2 \).
   - **Right-hand limit**: \( \lim_{x \to 2^+} f(x) = -2(2) + 2 = -4 + 2 = -2 \).
   - **Function value at \( x = 2 \)**: \( f(2) = -2(2) + 2 = -4 + 2 = -2 \).

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

### Conclusion:
The function is continuous for all real numbers. Thus, the answer to the question is:

**Yes, the function is continuous.**

Would you like a more detailed explanation or have any questions?

### Related Questions:
1. How would you find the derivative of this piecewise function?
2. What happens if the slopes of the two pieces are equal?
3. How can you determine the continuity of a function graphically?
4. Can a piecewise function have a jump discontinuity? If so, how can you identify it?
5. What is the difference between pointwise and interval continuity?

### Tip:
When analyzing piecewise functions for continuity, always check the left-hand and right-hand limits at the boundary points.

Graph the function f(x) defined as: f(x) = {x - 4 if x < 2, -2x + 2 if x >= 2}. Determine whether the function is continuous.

Learn how to graph the piecewise-defined function f(x) and determine if it's continuous. This problem covers key concepts of piecewise functions, including graphing linear pieces and checking continuity at a boundary point. Suitable for high school students.

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