What are some examples of non differentiable functions?

1 Answer
Mar 13, 2015

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