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To ensure the function \( f(x) \) is continuous across the entire real number line, we need to make sure it is continuous at \( x = 2 \). This requires the left-hand limit and the right-hand limit at \( x = 2 \) to be equal, as well as equal to the function value at that point.

The function is given by:
\[
f(x) = 
\begin{cases}
x^2, & \text{if } x \leq 2 \\
ax + 2, & \text{if } x > 2
\end{cases}
\]

### Step 1: Continuity Condition at \( x = 2 \)

To ensure continuity, the following must hold:
\[
\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2).
\]

**Left-hand limit:**
\[
\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^2 = 2^2 = 4.
\]

**Right-hand limit:**
\[
\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (ax + 2) = 2a + 2.
\]

**Function value at \( x = 2 \):**
\[
f(2) = 2^2 = 4.
\]

### Step 2: Equating the Limits
For continuity at \( x = 2 \), we set:
\[
2a + 2 = 4.
\]

### Step 3: Solve for \( a \)
\[
2a = 4 - 2,
\]
\[
2a = 2,
\]
\[
a = 1.
\]

Thus, the function will be continuous on the entire real number line if \( a = 1 \).

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

Here are some follow-up questions you might find interesting:
1. What would happen if \( a \) were a different value?
2. How do we determine continuity for piecewise functions in general?
3. Can the function \( f(x) \) be differentiable at \( x = 2 \) when \( a = 1 \)?
4. How would the graph of \( f(x) \) look around \( x = 2 \)?
5. What are the implications of a function being continuous on the real line?

**Tip:** When dealing with piecewise functions, always check the boundaries where the definition changes, as continuity and differentiability issues often arise there.

Find the value of parameter a for which the function f(x) is continuous on the entire real number line, where f(x) = x^2 if x ≤ 2 and f(x) = ax + 2 if x > 2.

Learn how to find the value of parameter a to ensure continuity of the piecewise function f(x), where f(x) = x^2 for x ≤ 2 and f(x) = ax + 2 for x > 2. This problem involves limits and the Continuity Theorem, suitable for students in Grades 11-12.

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