**Contents**

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.

- Heine continuity implies Cauchy continuity without the Axiom of Choice

On this page we state and prove that every Heine continuous real function is also Cauchy continuous. In our proof we do not use the Axiom of Choice.