Exercise 1891Topology › Topology of ℝⁿM1Added 3 Nov 2026
Baire category theorem for $\mathbb{R}^n$
#Baire#complete metric#category
Statement
State Baire's theorem for and explain its use for the existence of functions continuous nowhere differentiable.
Hint
is a complete metric space; the set of continuous functions is meager in the space of all functions, but the set of nowhere-differentiable ones is residual.
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Solution
Baire's theorem. In a complete metric space, a countable intersection of dense open sets is dense.
Application. In the space of continuous functions on with the uniform norm, the set of nowhere-differentiable functions is a dense (residual) — hence non-empty (in fact, "typical").
Conclusion. Baire's theorem forces the existence of highly irregular but continuous functions.