Increasing and Strictly increasing

A function is said to be increasing on $\displaystyle [a,b]$ if for every $\displaystyle x_{1} ,x_{2}\in(a,b)$ where $\displaystyle x_{2} > x_{1}$then $\displaystyle f(x_{2}) > f(x_{1})$

I read in a book that function $\displaystyle y = f(x)$ is increasing on $\displaystyle [a,b]$ if $\displaystyle f'(x)\geq 0$ on the interval $\displaystyle (a,b)$(but $\displaystyle f'(x)$ should not become zero on any sub-interval of $\displaystyle (a,b))$

This is given in standard texts

So what exactly does strictly increasing mean i.e when is a function said to be strictly increasing on the interval $\displaystyle [a,b]$.

The definition of strictly increasing was given as follows:

The function $\displaystyle y = f(x)$ is strictly increasing on $\displaystyle [a,b]$ if $\displaystyle f'(x)> 0$ on the interval $\displaystyle (a,b)$

A confusion gets created with regard to use of $\displaystyle \geq$sign and $\displaystyle >$ sign

So what exactly is the definition of increasing function and what exactly is the definition of strictly increasing function