Problem
Determine if the function \(f(x) = \frac{x^2 - 4}{x - 2}\) is continuous over the interval \([-2, 4]\).
Solution
Step 1 ：Step 1: We first find the domain of the function. The function is defined for all real numbers except at \(x = 2\), where the denominator is zero and the function is undefined.
Step 2 ：Step 2: We then rewrite the function \(f(x)\) as \(f(x) = (x + 2)\) for all \(x \neq 2\) by factoring the numerator and cancelling out the \((x - 2)\) terms. This function is defined and continuous for all real numbers.
Step 3 ：Step 3: Observe that the interval \([-2, 4]\) includes the point \(x = 2\) where the original function is not defined. Therefore, the function \(f(x) = \frac{x^2 - 4}{x - 2}\) is not continuous over the interval \([-2, 4]\).
