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2014-12-20T03:38:45+05:30

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We can differentiate a function with a modulus operator.

     y = absolute (x) =  | x |. 

The function is continuous at every point and is differentiable too at every point except at x = 0.

Some functions with a modulus operator may not be differentiable at some points.

       y =  | tan x |
         ie.,  y = tan x         0 <=  x  < =  Pi/2
                    = - tan x        - Pi/2  <= x <= 0  or , 
                       or  tan z,      z = -x,  0 <= z <= Pi/2

       at all points except at y= x = 0, this function is continuous and differentiable.

     at  x = 0  =y,
              dy/dx on x = 0+, is  sec^2 x = sec^2 0 = 1
                 dy/dx on x = 0- or z = 0+ ,  is    sec^2 z  = 1                 

         Hence, the function is differentiable at x = 0.

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