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We are tasked with finding the value of \( a \) such that the given function \( g(x) \) is continuous for all \( x \in \mathbb{R} \).

The function is defined as:
\[
g(x) = 
\begin{cases} 
\frac{x^3 - 7x + 6}{x - 2}, & x \neq 2 \\
a, & x = 2
\end{cases}
\]

### Step-by-step solution:

To ensure that the function is continuous at \( x = 2 \), we need to ensure that the left-hand limit, right-hand limit, and the value of the function at \( x = 2 \) are all equal. 

1. **Continuity condition**: 
   \[
   \lim_{x \to 2} g(x) = g(2) = a
   \]
   This means we need to calculate \( \lim_{x \to 2} \frac{x^3 - 7x + 6}{x - 2} \) and set it equal to \( a \).

2. **Simplifying the expression**:
   We first factor the numerator \( x^3 - 7x + 6 \). Using polynomial division or factoring methods, we factor it as:
   \[
   x^3 - 7x + 6 = (x - 2)(x^2 + 2x - 3)
   \]
   Now we can simplify the function for \( x \neq 2 \):
   \[
   \frac{x^3 - 7x + 6}{x - 2} = \frac{(x - 2)(x^2 + 2x - 3)}{x - 2} = x^2 + 2x - 3
   \]
   for \( x \neq 2 \).

3. **Evaluating the limit**:
   Now, find the limit as \( x \to 2 \) of the simplified expression \( x^2 + 2x - 3 \):
   \[
   \lim_{x \to 2} (x^2 + 2x - 3) = (2^2 + 2 \times 2 - 3) = 4 + 4 - 3 = 5
   \]

4. **Continuity requirement**:
   For \( g(x) \) to be continuous at \( x = 2 \), we require:
   \[
   a = 5
   \]

Thus, the value of \( a \) that makes the function continuous is \( \boxed{5} \).

---

Let me know if you'd like any further details or explanations. Here are five related questions:

1. How do you factor a cubic expression like \( x^3 - 7x + 6 \)?
2. What is the difference between a removable and a non-removable discontinuity?
3. How do you apply L'Hopital's Rule for limits involving indeterminate forms?
4. Can you generalize the method for finding limits of piecewise functions at points of potential discontinuity?
5. How does polynomial long division work for simplifying expressions?

**Tip:** Always check if a function has factors that cancel out indeterminate forms (like \( 0/0 \)) before evaluating limits.

Bestem a slik at følgende funksjon er kontinuerlig for alle x ∈ R: g(x) = { (x^3 - 7x + 6)/(x - 2), x ≠ 2; a, x = 2 }

This problem asks to determine the value of 'a' that ensures the function g(x) is continuous for all x in R. The solution involves factoring a cubic polynomial and calculating limits to satisfy the continuity condition at x = 2.

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