ickedifferentierbara

ickedifferentierbara is a term found in some Swedish-language mathematical writings to denote functions that are not differentiable at any point. In English-language mathematics, the standard term is nowhere differentiable. The concept describes functions whose rate of change cannot be approximated by a linear map at any point in their domain.

Formally, a function f defined on an interval I is nowhere differentiable if, for every x in I, the derivative f'(x) does not exist. This means there is no point at which f has a tangent or a linear approximation, highlighting a high level of irregularity.

A classic example is the Weierstrass function, introduced by Karl Weierstrass in 1872. It is continuous everywhere but differentiable nowhere for a suitable choice of parameters in its classical series representation, typically written as a sum of scaled cosine terms with increasing frequencies. This function demonstrates that a continuous function can exhibit extreme irregularity, challenging the intuition that continuity implies some degree of smoothness.

The study of nowhere differentiable functions is a key topic in real analysis and fractal geometry. They provide counterexamples to naïve regularity assumptions and help gauge the limits of differential calculus. In probability theory, stochastic processes such as Brownian motion yield almost surely nowhere differentiable sample paths, reinforcing the idea that irregular behavior is common in broader mathematical contexts.

In practice, ickedifferentierbara appears mainly as a translation variant in Swedish sources; the English literature typically uses nowhere differentiable or nowhere-differentiable to describe these functions.