I have some confusion in difference between monotone increasing function and Increasing function. For example
$$f(x)=x^3$$ is Monotone increasing i.e, if $$x_2 \gt x_1$$ then $$f(x_2) \gt f(x_1)$$ and some books give such functions as Strictly Increasing functions.
But if
$$f(x)= \begin{cases}
x & x\leq 1 \\
1 & 1\leq x\leq 2\\
x-1 & 2\leq x
\end{cases}
$$
Is this function Monotone increasing?
Best Answer
I'm used to the following: $f$ is
increasing if $x< y \Rightarrow f(x)\le f(y)$
strictly increasing if $x< y \Rightarrow f(x)< f(y)$
decreasing and strictly decreasing: similar, with the inequalities for $f$ reversed.
Monotonic: either increasing or decreasing
strictly monotonic: either strictly increasing or strictly decreasing.
In particular, monotonically increasing is the same as increasing, strictly monotonically increasing the same as strictly increasing.
(In general there is always some freedom when it comes to definitions, this is why I wrote 'I'm used to'. In case you are reading a textbook on analyisis the author should define these terms and then stick to them).