Final answer:
A continuous function maintains a constant sign within each specified interval because it cannot 'jump' from one value to another without passing through intermediate values, adhering to the Intermediate Value Theorem. Since the function is only undefined or zero at specific points, there cannot be a sign change within the intervals without contradicting these points. The continuous nature and finite limits of the function as it approaches these points further support the sign's constancy across the intervals.
Step-by-step explanation:
To prove that a continuous function f maintains a constant sign on each of the intervals given, we must consider the properties of continuous functions. By definition, a continuous function cannot 'jump' from one value to another without passing through all intermediate values. This is commonly known as the Intermediate Value Theorem. If the function f were to change signs between two points within an interval, say a and b, there must exist at least one point c in the interval (a, b) such that f(c) would be zero. However, we are given that f is undefined or zero only at the specific points aj. Thus, within each interval between these points, the sign of f must remain the same, because otherwise, there would be additional points where f is zero, which contradicts our initial information.
The properties of a continuous function also dictate that if the function has a finite limit as x approaches any of the given points from within an interval, it cannot jump to infinity or to an undefined state, further supporting the idea that the function's sign remains constant on each interval. The notion of sign includes both positive and negative (represented by +ve and -ve), which are always relative to zero. As a consequence of these rules and the continuous nature of f, one can conclude that f does not change sign on any of the open intervals specified.
It's also essential to consider the behavior of a function like y = 1/x, which illustrates asymptotic behavior. As x approaches zero, y approaches infinity, meaning the sign of y remains consistent, highlighting another aspect of continuous functions' behavior. This, along with the details previously discussed, allows us to confidently state the sign constancy of f on each of the intervals.