Final answer:
The function is discontinuous at a = 4 because it has a different value for x = 4 than for other values of x.
Step-by-step explanation:
The function f(x) = (x² - 4x)/(x² - 16) is a rational function with two linear factors. The domain of this function is the set of all real numbers except x = 4 and x = -4. At these two points, the denominator of the function is equal to zero, which means that the function is undefined at these points.
At x = 4, the numerator of the function is equal to zero, which implies that the function is continuous at this point. However, the value of the function at this point is equal to 1, which is different from its value at other points. Therefore, the function is discontinuous at x = 4.
To illustrate this, consider the function's graph. The graph shows that at x = 4, the function has a different value than it has at other points. This is because at x = 4, the function has a value of 1, whereas at other points, the function has a value of 0. Therefore, the function is discontinuous at x = 4.
In conclusion, the function f(x) = (x² - 4x)/(x² - 16) is discontinuous at a = 4 because it has a different value for x = 4 than for other values of x.