Final answer:
The function f(x) is continuous for any value of the constant c as long as the function remains a horizontal line over the entire real number line. Since a horizontal line is continuous by nature, option D) c can be any real number is correct, provided the function continues without breaks for all real numbers.
Step-by-step explanation:
The student is asking about the conditions under which the function f(x) is continuous over the entire real number line. According to the provided information, f(x) is a horizontal line over the interval 0 ≤ x ≤ 20. A horizontal line represents a constant function, meaning that f(x) has the same value for all x in the domain. To ensure continuity over the entire real number line, the function f(x) must not have any breaks or jumps. Since a horizontal line is continuous by nature, the constant c can be any real number to fulfill this condition, as long as f(x) also satisfies this outside the interval of 0 ≤ x ≤ 20.
Therefore, option D) c can be any real number is correct, provided the function maintains the horizontal line behavior for all real numbers. If there's a different behavior of f(x) beyond the interval mentioned, there is not enough information given to conclude the continuity based on the value of c.