Coming from the simple sine function (remember of Fourier series), German mathematician Karl Weierstrass became the first to publish an example of a continuous, nowhere
differentiable 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).
Click through for an example of this in R.