Reason for the diferentiability of function on open interval in Rolle's Theorem "Let $f(x)$ be a continuous function on $[a,b]$ and differentiable on $]a,b[$ ,further let $f(a)=f(b)$. Then there exist at least one number $c \in ]a,b[$ such that $f'(c)=0$". The above statement is the Rolle's Theorem. Why we take open interval for differentiability rather then close interval. What's the reason behind it??
See the proof of the theorem. If $f$ isn't constant, the point where $f$ reaches the max or the min $\in(a,b)$, so is a local extremum...
Simply put: there is no two-side derivative for the bounders of the closed interval. only one side derivative exists for $a$ and $b$. so we must exclude the two points.