Question

What are some examples of non differentiable functions?

Answer 1

There are three ways a function can be non-differentiable. We'll look at all 3 cases.

Case 1
A function in non-differentiable where it is discontinuous.

Example (1a) f`(x)=cotx` is non-differentiable at `x=n pi` for all integer `n`.

graph{y=cotx [-10, 10, -5, 5]}

Example (1b) `f(x)= (x^3-6x^2+9x)/(x^3-2x^2-3x)` is non-differentiable at `0` and at `3` and at `-1`
Note that `f(x)=(x(x-3)^2)/(x(x-3)(x+1))`
Unfortunately, the graphing utility does not show the holes at `(0, -3)` and `(3,0)`

graph{(x^3-6x^2+9x)/(x^3-2x^2-3x) [-10, 10, -5, 5]}

Example 1c) Define `f(x)` to be `0` if `x` is a rational number and `1` if `x` is irrational. The function is non-differentiable at all `x`.

Example 1d) description : Piecewise-defined functions my have discontiuities.

Case 2
A function is non-differentiable where it has a "cusp" or a "corner point".
This occurs at `a` if `f'(x)` is defined for all `x` near `a` (all `x` in an open interval containing `a`) except at `a`, but `lim_(xrarra^-)f'(x) != lim_(xrarra^+)f'(x)`. (Either because they exist but are unequal or because one or both fail to exist.)

Example 2a) `f(x)=abs(x-2)` Is non-differentiable at `2`.
(This function can also be written: `f(x)=sqrt(x^2-4x+4))`

graph{abs(x-2) [-3.86, 10.184, -3.45, 3.57]}

Example 2b) `f(x)=x+root(3)(x^2-2x+1)` Is non-differentiable at `1`.

graph{x+root(3)(x^2-2x+1) [-3.86, 10.184, -3.45, 3.57]}

Case 3

A function is non-differentiable at `a` if it has a vertical tangent line at `a`.
`f` has a vertical tangent line at `a` if `f` is continuous at `a` and

`lim_(xrarra)abs(f'(x))=oo`

Example 3a) `f(x)= 2+root(3)(x-3)` has vertical tangent line at `1`. And therefore is non-differentiable at `1`.

graph{2+(x-1)^(1/3) [-2.44, 4.487, -0.353, 3.11]}

Example 3b) For some functions, we only consider one-sided limts: `f(x)=sqrt(4-x^2)` has a vertical tangent line at `-2` and at `2`.

`lim_(xrarr2)abs(f'(x))` Does Not Exist, but

`lim_(xrarr2^-)abs(f'(x))=oo`

graph{sqrt(4-x^2) [-3.58, 4.213, -1.303, 2.592]}

Example 3c) `f(x)=root(3)(x^2)` has a cusp and a vertical tangent line at `0`.

graph{x^(2/3) [-8.18, 7.616, -2.776, 5.126]}

Here's a link you may find helpful:
http://socratic.org/calculus/derivatives/differentiable-vs-non-differentiable-functions