Extrema and nowhere differentiable functions (Q1089118)

Simple constructions of continuous nowhere differentiable functions f, g, h for which A) f has neither proper local maxima nor proper local minima, and \(f^{-1}(y)\) is, for every y, a perfect set (and f has neither finite nor infinite derivatives at any point); B) g has no proper maxima but has a dense set of proper minima; the set of functions with this property is of first category; C) h has proper local maxima and proper local minima, each on a dense set. The authors also show that if f is continuous and has no interval of monotonicity, then the sets where f has a local maximum (or minimum) are dense and of first category.