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?