Differentiability in an interval:

In open interval

A function f(x) is defined on an open interval (a, b) is differentiable or derivable in open interval (a, b) if it can be differentiated at each point of (a, b)

In close interval

A function f(x) can bedefined on [a, b] is said to be differentiable or derivable at end points a and b if it is differentiable from right at a and from left at b. Or we can say that both exist.

"If f can be derived in an open interval (a, b) and also at end points a and b, then f is said to be derivable in closed interval [a, b]".

For checking differentiability on the closed interval [a, b] we say,

"A function f is a differentiable function if it is differentiable at every point of the domain."

Example: Let  f(x)  = x³ - x² + x + 1

                        Discuss continuity and differentiability of g(x) in (0, 2).

Solution:       f(x) = x³ - x² + x + 1

                        => f'(x) = 3x² - 2x + 1 > 0 ∀ x

                        => f(x)  is increasing  function on [0, x]

                        Clearly g(x)  is continuous at  x = 1 and  not differentiable at x = 1. 

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