Question

Is a function that consists of a finite set of ordered pairs sometimes, always, or never continuous?

A) Sometimes
B) Always
C) Never
D) Depends on the nature of the ordered pairs

Answer 1

A function consisting of a finite set of ordered pairs is never continuous because it is discrete and only defined at specific points without connecting intervals.

Step-by-step explanation:

The question asks whether a function that consists of a finite set of ordered pairs is sometimes, always, or never continuous. Continuity in a function essentially means that you can draw the graph of the function without lifting your pencil from the paper. For a finite set of ordered pairs, the function defined by these pairs is not continuous because there are only isolated points with no connecting intervals. Hence, the most accurate answer is that such a function is never continuous, as there aren't intervals over which the function is defined.

To give an example, consider the function defined by the following ordered pairs: { (1, 2), (3, 4), (5, 6) }. This is clearly a discrete function as it only takes on specific values at these particular points and is undefined everywhere else. You cannot draw a continuous line through these points without defining the function for other values as well.