Article Details Article ID B.1957.49.31 Title A Note on Derivatives of a Function Author H.M. Sengupta & B.K. Lahiri Issue Vol. 49, No. 4, - 1957 Article No. 31, Pages 189-191
Abstract Recently Prof. Fort Jr. (1951) has proved a striking theorem regarding the differentiability of a function which is discontinuous over an everywhere dense set and continuous over an everywhere dense set. He has proved that if the set of points where the function is discontinuous be everywhere dense and if there be an everywhere dense set of points where f(x) is continuous, then the set of points (if it exists) where the function is differentiable is a set of the first category. He proves this by showing that the set of points where f(x) is continuous but not differentiable is a residual set. In this note it is a proposed to show that in case there is an everywhere dense set of points when f(x) is discontinuous and an everywhere dense set of points where f(x) is continuous, then there always exists a residual set at each point of which at least one of the four derivatives D^+f, D_+f, D^-f is infinite. In this connection, we refer to an article by W.H. Young (1903) [see Hobson, 1927] where it is proved that for any function f(x) defined in a
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