# 10. Prove that the greatest integer function defined by    is not differentiable atx = 1 and x =  2.

Given function is

We know that any function is differentiable when both
and         are finite and equal
Required condition for function to be differential at x  = 1  is

Now, Left-hand limit of the function at x = 1 is

Right-hand limit of the function at x = 1 is

Now, it is clear that
R.H.L. at x= 1    L.H.L. at x= 1  and L.H.L. is not finite as well
Therefore, function  is not differentiable at x = 1
Similary, for x = 2
Required condition for function to be differential at x  = 2  is

Now, Left-hand limit of the function at x = 2 is

Right-hand limit of the function at x = 1 is

Now, it is clear that
R.H.L. at x= 2    L.H.L. at x= 2  and L.H.L. is not finite as well
Therefore, function  is not differentiable at x = 2

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