why it is written continuous in closed interval and differentiable in open interval in rolle's and langrange's theorem.?

Functions are usually not considered to be differentiable at the end points of the domain. Thus the wording is as such.

Condition should satisfy the eqn. on applying Roll''s theorem and langranges!

thus it is!

In calculus, Rolle''s theorem essentially states that a differentiable function which attains equal values at two distinct points must have apoint somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero.

Standard version of the theorem 

If a real-valued function f is continuous on aclosed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists a c in the open interval (a, b) such that

This version of Rolle''s theorem is used to prove the mean value theorem, of which Rolle''s theorem is indeed a special case. It is also the basis for the proof of Taylor''s theorem.