# How do we find the set of points where a particular function is differentiable when we have to check for the entire domain of the function? How to choose the expected points of discontinuity?

42 Points
14 years ago

Dear Eshita,

Please note that for a function to be differentiable at a point we need the following conditions:

1. It should be defined and be continuous at that point.
2. It's left hand and right hand derivatives should exist and be equal

Now first the first part. Generally you shall be given a function as a composition of some functions that you know and this could be piecewise over the x-axis (say one composition for one part of x-axis and another for another part)

e.g. y = sin(ln(x)) for x<0 and cos(ln(x)) for x >= 0

First thing you should check where the functions are not defined. like ln(x) is not defined for x<=0, 1/x is not defined for x=0. See where you get these situations in your functions. The operands to these functions should not be coming to those values. Next check for continuity of those functions. Again you would know where a function is not continuous like step function is not continuous at x=0. Once you have these check where those functions were not differentiable themselves. Generally there are hardly any simple functions which are discontinuous at points where they are countinous. From simple I mean function like trigonometric, trigonometric inverses, exponentials, polynomials etc.

So basically compositions would now give you trouble mostly for boundary points. If boundary points are well defined and there is continuity then calculate RHD (right hand derivate) and LHD and see if they are equal. Now you are done :)

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