The blancmange function: Continuous everywhere but differentiable nowhere

Extract

One of the problems of the first introduction to the calculus and the subsequent mental imagery developed by the student is that the functions involved are usually given by simple formulae such as f(x) = xⁿ and the derivatives calculated by formulae crunching: f′(x) = nxⁿ⁻¹. The fundamental ideas of the calculus and any relational understanding recede into the background. Getting out of the strait-jacket and considering more general functions at some stage is rarely considered. When it is, it is usually performed in the context of university analysis where pictures are banned because they are claimed to mislead the intuition.