Continuous Nowhere Differentiable Function
Proof of the existence of a continuous function f:[0,1]->[0,1] that is nowhere differentiable (uses the Banach Contraction Principle).
Continuous Nowhere Monotonic Function
Proof of the existence of a continuous function f:[0,1]->|R that is neither increasing nor decreasing on any subinterval of [0,1] (uses the Baire Category Theorem).
These proofs have been adopted from the book Topological Spaces: From Distance to Neighborhood by Gerard Buskes and Arnoud van Rooij.See also