Example of a PDF that is not differentiable

Question

Suppose my continuous probability distribution has a PDF. I'm wondering if the PDF is differentiable on the real line. The definition does not require that a pdf be differentiable, but I don't recall seeing examples of pdfs that aren't differentiable. Can the pdf be non-differentiable?

Answer 1

Sure, just find any non differentiable, continuous, nonnegative function which integrates to $1$. For example $f(x)=x 1_{(0,1)}(x)+(2-x)1_{(1,2)}$.

Answer 2

Another very classical non-differentiable pdf is the Laplace distribution (or double exponential) defined by $$f(x)=\dfrac{1}{2}e^{-|x|}$$ (https://en.wikipedia.org/wiki/Laplace_distribution). This non-differentiability being limited to $x=0$.

Laplace distribution occurs for example in a natural way as the pdf or $X_1-X_2$ where $X_1,X_2$ are independant identically distributed exponential random variables (with pdf $e^{-x}\mathbb{1}_{[0,\infty)}$).