Difference between Increasing and Monotone increasing function 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? 

Let $y=f(x)$ be a differentiable function on an interval $(a,b)$. If for any two points $x_1,x_2 \in (a,b)$ such that $x1 \lt x2$, there holds the inequality $f(x_1) \leq f(x_2)$, the function is called increasing on ths interval. 
If there holds the inequality $f(x_1) \lt f(x_2)$, the function is called  strictly increasing on the interval.
