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( using L’Hospital Rule)
is equal to
Case 1 Let c be a real number which is not equal to any integer. for all real numbers close to c the value of the function is equal to [c]; i.e., . Also and hence the function is continuous at all real numbers not equal to integers.
Case 2 Let c be an integer. Then we can find a sufficiently small real number such that
This, in terms of limits mean that
Since these limits cannot be equal to each other for any c, the function is discontinuous at every integral point.
Therefore f'(x) exists at all
Further, f'(0) =