Final answer:
Continuous functions f and g on [a,b] that agree on all rationals must also agree on all reals in that interval due to continuity and the density of rationals. The result may vary for a subset A depending on its properties and the continuity definition used there.
Step-by-step explanation:
When we have two continuous functions on an interval [a,b], and they agree on all rational numbers in (a,b), to prove f(x) = g(x) for all x∈[a,b], we can use the fact that rational numbers are dense in the real numbers, and that any real number can be approached arbitrarily closely by a sequence of rational numbers. If f and g are continuous, then they must agree on the limit points of these sequences, which are the real numbers in the interval.
If f(r) = g(r) for all rationals r, and both f and g are continuous, then the limits of f and g as they approach any real number x must be the same. Thus, f(x) must equal g(x) for all x in [a,b]. Whether this result holds for an arbitrary subset A of ℝ depends on the properties of the subset and the definition of continuity on that subset.
If we apply this question to continuous probability functions, we can consider an example like f(x) being a probability density function over an interval [0, 20]. The probability that a value lies within a certain range is given by the area under the curve of f(x) over that range.