Question

In 2.2, recall that on an open interval, the end points are not included in the interval. For example, the open interval (2, 4) represents the real numbers between, but not including, 2 and 4. In a closed interval, the end points are included in the interval. For example, the closed interval [5,8] represents the real numbers between and including, 5 and 8. In some text books (including ours), they define increasing, decreasing, and constant on open intervals. Other texts do not require the intervals to be open, meaning they can be defined on closed intervals. What do you think? Should increasing, decreasing, and constant intervals be defined on open intervals or do you think they should be defined on closed intervals? Why?​

Answer 1

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Answer:

it depends

Explanation:

The ideas of "increasing" or "decreasing" have to do with the sign of the derivative of a function. The derivative of a function is a limit, which is only defined if the point can be approached from both sides. For a function that is only defined on an interval, the derivative is undefined (hence "increasing" or "decreasing" are undefined) at the end points of the interval.

When the function is defined on an interval, "increasing" or "decreasing" can only be determined on that open interval. There may also be critical points within an interval at which the derivative is either zero or undefined. Those points must also be excluded from any interval of "increasing" or "decreasing".

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If a function is defined over a domain that extends beyond the interval of interest, then the derivative may very wll be defined a the end points of the interval of interest. As a simple example, consider a line with defined non-zero slope: y = kx, k≠0. For k>0, the line will be increasing everywhere. The slope is defined at the end points of any finite interval, so the function can be said to be "increasing" on the closed interval.

Similarly, if the (finite) interval of interest includes the vertex of a parabola defined for all real numbers, the function will be "increasing" on one side of the vertex, and "decreasing" on the other side. Both the "increasing" and "decreasing" intervals will be half-open intervals. The point at the vertex will not be included in either of them.