Final answer:
Without the specific function, we can only refer to the general principles that linear functions are continuous for all real values of a and b. Hence, the function will be continuous regardless of the values of a and b.
Step-by-step explanation:
The question asks for what values of a and b is a certain function continuous at every x. A continuous function means that there are no gaps, jumps, or holes in the graph of the function. Analyzing the choices given (a, b, c, d), without the specific function provided, we can refer to the principles of continuity and the given context clues.
In general, if the function mentioned is a linear function, such as y = a + bx, then the function will be continuous for all real numbers provided a and b are real numbers, regardless of their values. This includes a = 0, b = 0, or both being nonzero. This is because the graph of a linear function, as per the information provided in Figure 12.4, will always be a straight line, which is inherently continuous.
However, since the provided references do not give a specific function or describe the nature of a and b's roles in relation to continuity in detail, it is impossible to give a definitive answer on whether a or b (or both) should be zero or not.
Therefore, based on standard knowledge regarding linear functions and the context clues, the most likely answer is that the function will be continuous for any real values of a and b, which implies the MCQ answer could be none of the above as all given options suggest restrictions on a or b, which are unnecessary for a linear function to be continuous.