Let $$f: \mathbb{R} \to \mathbb{R}$$ be a continuous function such that $$f(3x) - f(x) = x$$. If $$f(8) = 7$$, then $$f(14)$$ is equal to:

We have $$f: \mathbb{R} \to \mathbb{R}$$ continuous, with $$f(3x) - f(x) = x$$, $$f(8) = 7$$.

Let $$f(x) = ax + b$$. Then:

$$f(3x) - f(x) = 3ax + b - ax - b = 2ax = x$$

$$\implies a = \frac{1}{2}$$

So $$f(x) = \frac{x}{2} + b$$.

$$f(8) = \frac{8}{2} + b = 4 + b = 7 \implies b = 3$$

$$f(x) = \frac{x}{2} + 3$$

$$f(14) = \frac{14}{2} + 3 = 7 + 3 = 10$$

Verification: $$f(3 \times 14) - f(14) = f(42) - f(14) = (21 + 3) - (7 + 3) = 24 - 10 = 14$$ ✓

Therefore, the correct answer is Option B: 10.