In Rolle’s theorem why we take continuity in close interval and differentiability in open interval?

Nice question, Maulik, I was searching the net for the same question and it's answer, here is what I understand, take example of derivative, derivative by definition is:

(f(x+h) – f(x))/h here h can be positive or negative for the term f(x +h)

to make sense or to be valid, the function f must be defined “around “ the point x, this mathematically define

a function in an open interval.

Hope this helps, although 4 hours have passed since you asked :)