Smooth function

A function for which each value of the argument is a smooth point (cf. Smooth point of a function). A smooth function can be discontinuous. If a smooth function is continuous on an interval, the set of its points of differentiability is dense in the interval and has the cardinality of the continuum. There exist continuous smooth functions on the real axis that are not almost-everywhere differentiable. A smooth function has a derivative at each point of local extremum; as a result, the basic theorems of differential calculus, the theorems of Rolle, Lagrange, Cauchy, Darboux, etc., remain valid for smooth continuous functions.

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Notice that any additive function $f$ (i.e. $f(x+y)=f(x)+f(y)$ for all $x$ and $y$) is smooth. There exist additive functions that are continuous at no point.

The notion of a smooth function as introduced above is a rather uncommon one. Usually "smooth function" means "sufficient often differentiable function", most often $C^\infty$-function (infinitely often differentiable function); it can also mean "having modulus of smoothness satisfying certain growth conditions" (cf. also Smoothness, modulus of).