Question

Why a function is continuous in closed interval and differentiable on open interval?

Answer 1

r intervals do not yield the same properties tocontinuous functions defined on them. ..K

When we are studying a function in an interval [a,b], we should only be concerned about the interval [a,b]. It should not include any analysis of (a-h) or (b+h) where h→0.

When we analyze the continuity of a function at a point, we are concerned only with THAT point.

But differentiation deals with values slightly greater as well as slightly smaller than the point at which it is analyzed.

Like if you want to study the differentiability of a function at a point x=a, it would depend on the nature of the function at x=(a+h) as well as x=(a-h). But as mentioned earlier (in italics), we must not be concerned about x=(a-h). So we don't analyse the differentiability of a function at extreme points of the interval where the function is being studied.