# Is there a function that doesn't have a derivative?

I was wondering if such a function exist. I'm comfortable with derivatives of polynomial functions, and some other basic functions, but I'm wondering if there could exist a very complicated function that doesn't have a derivative.

• Google "Weierstrass function". – David Mitra Oct 3 '15 at 11:49
• Or, more simply, the absolute value function $|x|$ at $x=0$. – joriki Oct 3 '15 at 11:53
• Related: Are Continuous Functions Always Differentiable? (And other posts linked there might be also of interest.) – Martin Sleziak Oct 3 '15 at 13:01
• @mick, if my answer was useful, consider accepting it by clicking the gray tick on the left of it. – dbanet Oct 19 '15 at 18:08

The commonly known example is the Weierstrass Function $f$, defined as $$f(x)=\sum\limits_{k=1}^\infty\frac{\sin\left(\pi{k}^2x\right)}{\pi{k}^2}.$$

The $y=f(x)$ graph looks like this (and intuitively shows why it is differentiable nowhere): Another example would be the Dirichlet Function $D$, defined as $$D(x)=\begin{cases}1,\;\;x\in\mathbb{Q},\\0,\;\;x\in\mathbb{I}.\end{cases}$$

Its graph $y=D(x)$ would look like a pair of lines $\displaystyle{y=\frac{1}{2}\pm\frac{1}{2}}$ (which of cousre is not a graph of any function), so is uninteresting to show. The interesting part is that $D$ is actually discontinuous everywhere, and therefore differentiable nowhere.

• It's worth mentioning that the Weierstrass function is continuous everywhere. – joriki Oct 3 '15 at 12:10
• does the Weierstrass function pass the vertical line test? – set5 Oct 3 '15 at 13:05
• @mick Hint: It's a function. – Workaholic Oct 3 '15 at 13:18
• @mick, any function passes the vertical line test. If you meant to ask about the Dirichlet Function $D$ instead, the source of confusion is clear: its graph only looks like a pair of parallel lines, but actually for every $x_0$ the line $x=x_0$ crosses only one of the “lines” of the graph (and “goes through a hole in the second one”, if that adds intuition). – dbanet Oct 3 '15 at 13:18