Is Non-Increasing the Same as Decreasing?

[ozkan12]

Can we say non-increasing equivalent to decreasing (For sequence or functions) ? İn some book, I see that...Can you say something ?

 

[Ackbach]

Non-increasing includes the category of constant, whereas I would think many authors would not allow that for a decreasing function.

It's much like the following:
\begin{align*}
\text{Non-negative integers}&=\{0,1,2,\dots\} \\
\text{Positive integers}&=\{1,2,3,\dots\}
\end{align*}

 

[ozkan12]

I didnt understand dear Ackbach...What I wonder is non-increasing equivalent to decreasing ?

 

[Ackbach]

They're not equivalent in my mind. A horizontal line like $y=f(x)=2$ is non-increasing. That is, it's not getting bigger. But it's also not decreasing - getting smaller. So a horizontal line would be in the set of non-increasing functions, but it would not be in the set of decreasing functions. So the set of non-increasing functions is not equal to the set of decreasing functions. You could define these two sets this way:
\begin{align*}
\text{non-increasing functions}&=\{f|(x_1<x_2) \implies (f(x_2)\le f(x_1))\} \\
\text{decreasing functions}&=\{f|(x_1<x_2)\implies (f(x_2)< f(x_1)) \}.
\end{align*}
One has strict inequality - the decreasing functions - and one does not.

 

[ozkan12]

İn some sources, I see that if f function is non increasing, then f is decreasing.

https://en.wikipedia.org/wiki/Monotonic_function

İf knowledges in this souce is true, can we say same thing for sequence ?

 

[Ackbach]

İn some sources, I see that if f function is non increasing, then f is decreasing.

https://en.wikipedia.org/wiki/Monotonic_function

İf knowledges in this souce is true, can we say same thing for sequence ?

This is strictly by author. You have to know how the author you're reading defines these terms. It's the true the Wiki seems to define non-increasing and decreasing as the same thing. I would find that confusing. I know other authors do not define it the same way. I'm afraid I'm not able to say much beyond this, other than an author would probably use the same sort of definition for sequences as for functions.

 

[ozkan12]

This is very confusing for me...I search these definitions but there is many definitions but they have different properties...I don't know what I do ? Thank you for your attention dear...

 

[Ackbach]

This is very confusing for me...I search these definitions but there is many definitions but they have different properties...I don't know what I do ? Thank you for your attention dear...

You have to interpret in context. You work in someone's book like Kirkwood or Rudin, you find their definition, and you go with that while you're working in that book. What book are you using? And how does that author define increasing or decreasing functions? And how does the word "monotone" fit in?

 

[ozkan12]

I work on some analysis books and on İnternet resources...But as I say, I found different things..

 

[Ackbach]

I work on some analysis books and on İnternet resources...But as I say, I found different things..

This isn't a problem. When you're reading one author, always use his definition. When reading another author, use his definition. Don't mix and match, and you'll be fine.