Write some examples of non differentiable functions

There are some cases when a function can be non-differentiable.
- Firstly, a function in non-differentiable where it is discontinuous.
For example, ${\displaystyle f\left(x\right)=\cot x}$ is non-differentiable at ${\displaystyle x=n}$ ${\displaystyle \pi }$ for all integer ${\displaystyle n}$.
Thus, the graph is ${\displaystyle \{y=\cot x[-10,10,-5,5]\}}$
Another example: ${\displaystyle f\left(x\right)=\frac{(x^3-6x^2+9x)}{(x^3-2x^2-3x)}}$ is non-differentiable at 0 and at 3 and at -1.
Note that ${\displaystyle f\left(x\right)=\frac{\left(x{(x-3)}^2\right)}{\left(x(x-3)(x+1)\right)}}$
And the graph: ${\displaystyle \left\{\frac{x^3-6x^2+9x}{x^3-2x^2-3x}[-10,10,-5,5]\right\}}$ If we define ${\displaystyle f\left(x\right)}$ to be ${\displaystyle 0}$ if x is a rational number and ${\displaystyle 1}$ if ${\displaystyle x}$ is irrational, the function is non-differentiable at all x.
- A function is non-differentiable where it has a "cusp" or a "corner point".
If ${\displaystyle f^{\prime }\left(x\right)}$ is defined for all ${\displaystyle x}$ near ${\displaystyle a}$ (all x in an open interval containing a) except at ${\displaystyle a}$, but ${\displaystyle \underset{x\to a^{-}}{lim}{f}^{\prime }\left(x\right)\ne lim\{x\to a^{+}\}{f}^{\prime }\left(x\right)}$.
Another example: ${\displaystyle f\left(x\right)=\left|x-2\right|}$. It is non-differentiable at ${\displaystyle 2}$. (This function can also be written as ${\displaystyle f\left(x\right)=\sqrt{(x^2-4x+4)}}$
So, the graph: ${\displaystyle \left\{\left|x-2\right|[-3.86,10.184,-3.45,3.57]\right\}}$
Example: $f(x)=x+\sqrt[3]{(x^2-2x+1)}$ Is non-differentiable at ${\displaystyle 1}$.
Graph: $x+\sqrt[3]{(x^2-2x+1)}[-3.86,10.184,-3.45,3.57]$
Finally, a function is non-differentiable at a if it has a vertical tangent line at a.
For example: $f(x)=2+\sqrt[3]{(x-3)}$ has vertical tangent line at 1. And therefore is non-differentiable at 1.
Graph: ${\displaystyle \{2+{(x-1)}^{\frac{1}{3}}[-2.44,4.487,-0.353,3.11]\}}$
For some functions, we only consider one-sided limts: ${\displaystyle f\left(x\right)=\sqrt{(4-x^2)}}$ has a vertical tangent line at ${\displaystyle -2}$ and at ${\displaystyle 2}$.
${\displaystyle \underset{x\to 2}{lim}\left|f^{\prime }\left(x\right)\right|}$ does not exist, but