Topological Methods In Real Analysis

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.

See also

• 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.