Is it true that $f(x)=k ,k\in \mathbb{R}$ is a decreasing and increasing and constant function in the same time over it's domain of definition?

Question

It is known that $f(x)=k,k\in \mathbb{R}$ is a constant function ,one of my friend argued me that he has an example of functions which it is increasing and decreasing and constant in the same time over its domain of definition such that he gave me the following example $f(4)=5 $, I said him that is just a constant function and we can't say that is satisfies monotonicity namely increasing and decreasing and constant in the same time since that is needed to be compared with other point , Any comment?

Answer 1

It depends on the definitions actually. If for increasing and decreasing he means "weakly", then he is correct. Recall a function $f:\mathbb{R} \rightarrow \mathbb{R}$ is called increasing(decreasing) if when $x<y$ then $f(x) \le f(y)$ ($f(x) \ge f(y)$). So a constant function is both increasing than decreasing. Of course it is not strictly increasing or strictly decreasing.

A same thing can happen with convexity. You may say that a linear function like $f(x)=x$ is both convex and concave, indeed it is linear.

Answer 2

Your 'friend' is using extremely sloppy notation (or you are sloppy in stating what he said). The proper counter-example to your false claim is that any real function on a singleton set of reals is constant and (strictly) increasing and (strictly) decreasing, and this is a trivial basic mathematical fact.