One of the more famous applications of the Baire category theorem, as first
One of the more famous applications of the Baire category theorem, as first applied by Banach and Mazurkiewicz, is that most functions on the closed interval zero one, with some restrictions, do not have this property anywhere. The "classes" of this property are represented by a C followed by a superscript number, C superscript infinity corresponding to a smooth function. If a function is monotone on an open interval, then it has this property almost everywhere on that interval according to (*) Lebesgue's (luh-beg's) theorem. A function possessing this property is a necessary precondition for applying Taylor's theorem. As exemplified by the Weierstrass function, a necessary but not sufficient condition for this property is that a function be continuous. For 10 points, identify this property possessed by functions whose derivative exists at every point in their domain.
One of the more famous applications of the Baire category theorem, as first applied by Banach and Mazurkiewicz, is that most functions on the closed interval zero one, with some restrictions, do not have this property anywhere.
differentiable [or word forms, such as differentiability; or analytic functions; or obvious equivalents, such as the derivative exists until it is read]