Fractal curves like the Koch snowflake are likewise continuous and nowhere differentiable. John McCarthy (yes, the inventor of Lisp) has a simpler to analyze c

Fractal curves like the Koch snowflake are likewise continuous and nowhere differentiable. John McCarthy (yes, the inventor of Lisp) has a simpler to analyze construction of a Weierstrass-like function from a triangle wave: http://www-formal.stanford.edu/jmc/weierstrass/weierstrass.html http://www-formal.stanford.edu/jmc/weierstrass/weierstrass.h... Here's the big picture. When you analyze variations in f at the kth scale by setting |dx| = 2^(-2^k), the only term that really contributes is the kth term, 2^(-k) g(2^(2^k) x), since the higher terms completely vanish and the lower terms are insignificant by comparison. Thus a difference quotient df/dx at the kth scale is very nearly the derivative of 2^(-k) g(2^(2^k) x), which is just +/- 2^(-k) 2^(2^k). This has unbounded magnitude as k increases (the exponential term 2^k outgrows the linear term -k). For a differentiable function the rate of variation should settle down at finer scales. Instead of settling down at finer scales, McCarthy's function actually gets progressively more jagged. If you want to grok the details, step 3 in his proof may be too terse. When bounding max(n < k) |dg(2^(2^n) x)|, he uses that the greatest variation in g occurs when x and x+dx are on the same line segment. The max slope of any g(2^(2^n) x) for n < k occurs when n = k-1 and is +/- 2^(2^(k-1)). Hence max(n < k) |dg(2^(2^n) x)| <= 2^(2^(k-1)) |dx| = 2^(2^(k-1)) 2^(-2^k).