{"history":[],"article":{"id":"0160261d-7797-45f2-9dc1-8865bb8fe7b9","title":"Weierstrass function","slug":"weierstrass-function","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **Weierstrass [function](https://bliptext.com/articles/function-mathematics)**, named after its discoverer, [Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass), is an example of a real-valued [function](https://bliptext.com/articles/function-mathematics) that is [continuous](https://bliptext.com/articles/continuous-function) everywhere but [differentiable](https://bliptext.com/articles/differentiable-function) nowhere.\nIt is also an example of a [fractal curve](https://bliptext.com/articles/fractal-curve).\nThe Weierstrass function has been historically served the role of a [pathological](https://bliptext.com/articles/pathological-mathematics) 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.\nAt least two researchers formulated continuous, nowhere differentiable functions before Weierstrass, but their findings were not published in their lifetimes.\nAround 1831, [Bernard Bolzano](https://bliptext.com/articles/bernard-bolzano) (1781–1848), a Czech mathematician, philosopher, and Catholic priest, constructed such a function; however, it was not published until 1922.\nSee:\nAround 1860, Charles Cellérier (1818 - 1889), a professor of mathematics, mechanics, astronomy, and physical geography at the University of Geneva, Switzerland, independently formulated a continuous, nowhere differentiable function that closely resembles Weierstrass's function.\nCellérier's discovery was, however, published posthumously:\n# [Plot of Weierstrass function over the interval [−2, 2]. Like some other fractals, the function exhibits self-similarity: every zoom (red circle) is similar to the global plot. | upload.wikimedia.org/wikipedia/commons/6/60/WeierstrassFunction.svg]\n\n# Construction\nIn Weierstrass's original paper, the function was defined as a [Fourier series](https://bliptext.com/articles/fourier-series):\nwhere 0

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