The function presented in the question is:

$f(x) = \frac{x^2 - 6x + 3}{x - 6}$

To determine the domain on which this function is continuous, we need to consider where it is not continuous, which typically occurs when the denominator equals zero, making the function undefined.

In this case, the denominator is $x - 6$, so the function will be undefined when:

$x - 6 = 0 \implies x = 6$

Thus, the function is not continuous at $x = 6$. Therefore, the function is continuous for all real numbers except $x = 6$.

Domain: $(-\infty, 6) \cup (6, \infty)$

Would you like further details or have any questions? Here are some related questions to expand on this topic:

1. How do you find discontinuities for rational functions in general?
2. Can a function be continuous at a point where the denominator is zero?
3. How does the factorization of the numerator affect the continuity of the function?
4. What is the difference between a removable and non-removable discontinuity?
5. How can limits help us determine the behavior of a function near discontinuities?

Tip: Always check both the numerator and denominator when determining where a rational function is continuous.