Question

Explain why the function f(x) = 1/x is not strictly monotone on the set of non-zero real numbers.

I know that for a function to be strictly monotone, it must be either strictly increasing on a subset S of the domain D or strictly decreasing on a subset S of the domain D. In this case, f(x) is strictly decreasing on the interval (0, inf). Isn't f(x) also strictly decreasing on the interval (-inf, 0) also? I'm not seeing the issue. Is it because f(x) isn't continuous at x=0 ?

Answer 1

Look at the values of f(-1) and f(1)

Then look at the values of f(1) and f(2)

Answer 2

For a function to be strictly monotonic, it must be either strictly increasing or strictly decreasing. That is :

For all x < y , we must have f(x) < f(y)

or


For all x < y , we must have f(x) > f(y)

Answer 3

The given function satisfies neither

Answer 4

Oh okay. I guess I was considering the two intervals: (-inf, 0) and (0, inf) separately. Thank you! @ganeshie8

Answer 5

I see.. it is indeed strictly monotonic if we consider those intervals separately and constrain the domain to one of them

Answer 6

I see.. it is indeed strictly monotonic if we consider those intervals separately and constrain the domain to one of them. But if the domain is non-zero real numbers, clearly it is not monotonic..

Answer 7

That makes sense. Thanks again. @ganeshie8