# Little useless-useful R functions – Continuous, nowhere differentiable Weierstrass function

**R – TomazTsql**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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Coming from the simple sine function (remember of Fourier series), German

xs <- seq(-2*pi,2*pi,pi/100) plot(xs,sin(2*xs),type="l",ylim=c(-1,1)); abline(h=0,lty=3)

mathematician Karl Weierstrass became the first to publish an example of a **continuous, nowheredifferentiable function**. Weierstrass function (originally defined as a Fourier series) was the first instance in which the idea that a continuous function must be differentiable was introduced. This is an example of a fractal in a function (known as a fractal function) and also another of pathological functions (runs counter to some intuition).

The mathematical function is represented as:

Let b be a real number such that 0 < b < 1 and let a be a positive odd integer.

If ab > 1 and 2 3 > π ab−1 , then

is continuous on R and is not differentiable at any point in R.

And the R code:

weierstrass_curve <- function(x,a,b) { values <- 0 for (n in 0:100) { values <- values + (a**n * cos(b**n * pi * x)) } return(values) } len <- 1000 x <- seq(-2.4, 2.4, length.out=len) y <- weierstrass_curve(x,0.3,5) plot(x, y, type = "l", col = "red", main = "Weierstrass curve")

As always, the complete code is available on GitHub in Useless_R_function repository. The sample file in this repository is here (filename: *Weierstrass_function.R*). Check the repository for future updates.

Happy R-coding and stay healthy!

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**R – TomazTsql**.

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