# Functions which are continuous but non differentiable at a point except modulus function

I have noticed one thing during solving problems : That is, wherever I find a function which is continuous but non differentiable at a point, there has always been some |.|(modulus) function or [.](greatest integer function) in it.

I feel that's the only way one can have a sharp point in the graph of a function. Every other function tend to be smooth at all points.I am unable to manipulate them into a non differentiable continuous function by adding, multiplying, squaring or any other operations.

Is my assumption true ? If not, can you give an example of a function which is continuous but non differentiable at a point except modulus function or GIF function ?

• Try taking fractional roots. – David Mitra Oct 3 '15 at 10:49
• i.stack.imgur.com/saC8D.gif – ThePortakal Oct 3 '15 at 10:50
• $\sqrt{x}$ at $x=0$. – Michael Hoppe Oct 3 '15 at 10:51
• √x is actually √|x|. So, you are using |.| @MichaelHoppe – Shubham Oct 3 '15 at 10:54
• I guess, in your graph you have used more than one function: from infinity to zero you have a function and after zero you have a different function, I am talking about one single function @ThePortakal – Shubham Oct 3 '15 at 10:59

Consider the Thomae function; $f:[0,1]\to\mathbb R$ where $$f(x)=\begin{cases} \frac1q,& x\in\mathbb Q,\ x=\frac pq, \ p,q\in\mathbb Z, \ p\not\lvert q, \ q>0\\ 0,& x\notin\mathbb Q. \end{cases}$$ Then $f$ is continuous on $[0,1]\setminus\mathbb Q$ but not continuous on $[0,1]\cap\mathbb Q$. See this MSE question for a proof: Proving Thomae's function is nowhere differentiable.
For an even more pathological example, we have the Weierstrass function $w:\mathbb R\to\mathbb R$, defined by $$w(x) = \sum_{n=0}^\infty a^n\cos(b^n\pi x),$$ where $0<a<1$, $b$ is a positive odd integer, and $$ab > 1 + \frac32\pi.$$ This function is continuous everywhere but differentiable nowhere. The proof of this is not trivial; see this paper: https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf
Consider $f(x)=max(x, x^2)$.It is continuous but not differentiable at x=1.