= Generic nowhere-monotone continuous function
In $C([0,1])$ with the uniform norm, the continuous functions that are monotone on some interval of positive length form a <meagre set>. Indeed, it is enough to use intervals with rational endpoints. For each such interval, the nondecreasing and nonincreasing functions form closed sets with empty interior: a sufficiently small local triangular perturbation breaks the relevant inequality. The <Baire category theorem> therefore shows that a dense set of continuous functions is monotone on no nontrivial interval.
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