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Based on my book

The rate of change of $f(u)$ as $u$ changes can be measured by looking at the change in $f$ between two values of $u$, divided by the change in $u$: $$\frac{Δf}{Δu} = \frac{f(u_2)-f(u_1)}{u_2-u_1}$$For a monotonic transformation, $f(u_2)-f(u_1)$ always has the same sign as $u_2-u_1$. Thus a monotonic function always has a positive rate of change. This means that the graph of a monotonic function will always have a positive slope, as depicted in Figure 4.1A.monotonic and non monotonic graphs

Now what I don't understand is if the monotonic function is as simple as $f(u) = u +1$ (Changing it slightly from the non-monotonic function) how come it comes it will always have a positive rate of change compared to the non-monotonic function?

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  • $\begingroup$ 1. What does "normal function" mean in this context. 2. Do you know about the Cantor function? $\endgroup$ Commented Feb 3, 2019 at 14:18
  • $\begingroup$ @kimchilover Thank you very much for your quick reply, by normal function I mean the non monotonic one (so in my case $f(u) = u$) $\endgroup$
    – Fozoro
    Commented Feb 3, 2019 at 14:19
  • $\begingroup$ I still really don't know what you are asking. That everything is called $f$ is no help. $\endgroup$ Commented Feb 3, 2019 at 15:02
  • $\begingroup$ Unless I'm blatantly missing something, this is an odd definition of monotonic. Usually a function that only decreases would be considered monotonic too. But there might be debate about whether it's allowed to have sections where it stays constant. $\endgroup$
    – timtfj
    Commented Feb 3, 2019 at 15:25
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    $\begingroup$ I wouldn't trust that book very much if I were you, since that excerpt is quite misleading. The slope isn't $\Delta f/\Delta u$, it's the limit of that thing as $\Delta u \to 0$. And even if $f$ is strictly increasing, there may be points where the slope is zero, like for $f(u)=u^3$ at $u=0$. $\endgroup$ Commented Feb 3, 2019 at 15:49

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