Weierstrass function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Weierstrass function, named after its discoverer, <a href="/facts/Karl_Weierstrass/4U8vdQzL">Karl Weierstrass</a>, is an example of a real-valued <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> everywhere but <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a> nowhere. It is also an example of a <a href="/facts/Fractal_curve/jjanzwda">fractal curve</a>.
The Weierstrass function has historically served the role of a <a href="/facts/Pathological_(mathematics)/KmKHjJ1H">pathological</a> function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of <a href="/facts/Smoothness/mffL4ch2">smoothness</a>. These types of functions were disliked by contemporaries: <a href="/facts/Charles_Hermite/kj1bGlMr">Charles Hermite</a>, on finding that one class of function he was working on had such a property, described it as a "lamentable scourge"[disputed – discuss]. The functions were difficult to visualize until the arrival of computers in the next century, and the results did not gain wide acceptance until practical applications such as models of <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a> necessitated infinitely jagged functions (nowadays known as fractal curves).

Weierstrass function open-in-new

Weierstrass function