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Question

How do you know if a function is continuous and differentiable?


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Solution

Definition of Continuity :

A function is said to be continuous at a point x=a, if

limf(x)x exists, and

limf(x)x=f(a)

It implies that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=aexists and these parameters are equal to each other, then the function fis said to be continuous at x=a.

If the function is undefined or does not exist at a point, then we say that the function is discontinuous.

Definition of Differentiability :

f(x) is said to be differentiable at the point x=a if the derivativef'(x) exists at every point in its domain. It is given by

f'(a)=limx0f(a+h)-f(a)h

Hence, for a function to be differentiable at any point x=a in its domain, it must be continuous at that particular point but vice-versa is not always true.


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