Suppose $f$ is a derivable function defined over $[a,b]$. Then the following hold :
For a point $c$ in $[a,b]$, if $f'(c)>0$ then $f$ is increasing at $c$.
Moreover, if the derived function $f'$ is continuous at $c$, then there is a neighborhood of $c$ where $f$ is strictly increasing.
If $f'(x)≥0$ for all $x$ in $[a,b]$, then $f$ is increasing on $[a,b]$.
Are my conclusions correct?