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The function given is defined as a piecewise function:

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

To determine whether this function is continuous, we need to check if the function is continuous at the point where the pieces meet, which is at \( x = 2 \).

### Steps to check continuity at \( x = 2 \):

1. **Evaluate the left-hand limit (as \( x \) approaches 2 from the left)**:
   \[
   \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^2 = 2^2 = 4
   \]

2. **Evaluate the right-hand limit (as \( x \) approaches 2 from the right)**:
   \[
   \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (2x - 2) = 2(2) - 2 = 4 - 2 = 2
   \]

3. **Evaluate the function at \( x = 2 \)**:
   \[
   f(2) = 2^2 = 4
   \]

### Conclusion:
- The left-hand limit is \( 4 \).
- The right-hand limit is \( 2 \).
- The value of the function at \( x = 2 \) is \( 4 \).

For the function to be continuous at \( x = 2 \), the left-hand limit, the right-hand limit, and the function value at that point must all be equal. Here, since the right-hand limit (\( 2 \)) is not equal to the left-hand limit (\( 4 \)) or \( f(2) \), the function is **not continuous** at \( x = 2 \).

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

Here are five related questions for further exploration:
1. How do you check the continuity of a function at a given point in general?
2. What are the conditions for a function to be continuous at a point?
3. Can a function be continuous everywhere except at one point?
4. How does the continuity of a piecewise function depend on its pieces?
5. What is the difference between continuity and differentiability?

**Tip:** When dealing with piecewise functions, always check the limits from both sides at the points where the function changes its definition.

Learn how to determine the continuity of a piecewise function with a step-by-step explanation. This problem involves checking continuity at a point where the function changes its definition, covering concepts of limits and piecewise functions. Suitable for high school students.

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