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I was attempting a quiz in which I encountered this question, Now the graph of the floor function is constant in the interval $[0,1)$ so technically the function should be neither increasing nor decreasing, but the answer to the question is its both increasing and decreasing. So is the answer wrong or am I unaware of a concept?

Similarly, there was another question about the function $f(x)=x^2 -3x +2$ in the interval $[0,inf]$ but the graph of the function comes down from 0 and there is a critical point at 1 and then goes up, so is decreasing and increasing in the interval $[0,inf)$ but the answer is its neither increasing nor decreasing, now one of my classmates gave me a line of reasoning that $f'(x)<=0 and f'(x)>=0$ in the inerval $[0,inf)$ which implies $f'(x)=0$ but I'm not really convinced with the argument because the graph clearly shows that the function is decreasing then increasing. So am I missing a concept here or the answers are wrong?

I tried to understand the arguments given in Can a function be increasing or decreasing at a point? this question but couldn't really make the comparisons, can someone help me here? Or suggest me some text to clarify this?

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  • $\begingroup$ The floor function on $[0,1)$ is identically $0$. What is your definition of increasing or decreasing? Are you going with strict increase/decrease or weak/monotone increase/decrease? If its the latter then the floor function is both increasing and decreasing (weakly/monotonically), but if its the former then the floor function is neither increasing nor decreasing (strictly). $\endgroup$
    – Dave
    Commented Apr 17, 2018 at 4:52

1 Answer 1

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increasing function: $x>y$ $\implies$ $f(x)\ge f(y)$

decreasing function: $x>y$ $\implies$ $f(x)\le f(y)$

strictly increasing function: $x>y$ $\implies$ $f(x)> f(y)$

strictly decreasing function: $x>y$ $\implies$ $f(x)< f(y)$

A constant function is increasing and decreasing. But it is neither strictly increasing nor strictly decreasing.

For $f(x)=x^2-3x+2$, it is increasing on $[1.5,\infty)$ and decreasing on $[0,1.5]$. But it is neither increasing nor decreasing on $[0,\infty)$.

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  • $\begingroup$ Should the second and fourth definitions say "decreasing" rather than "increasing"? $\endgroup$
    – Dave
    Commented Apr 17, 2018 at 4:55
  • $\begingroup$ *"decreasing" and "strictly decreasing" for the second and fourth rows. $\endgroup$
    – Jungleshrimp
    Commented Apr 17, 2018 at 4:55
  • $\begingroup$ Amended. Thanks. $\endgroup$
    – CY Aries
    Commented Apr 17, 2018 at 4:59
  • $\begingroup$ Okay, yes. I guess I mixed up neither increasing nor decreasing and a constant function. Thank you for clearing my doubt. CYAries and @Dave $\endgroup$
    – Sulaiman Qizilbash
    Commented Apr 17, 2018 at 5:01

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