What are some examples of non differentiable functions?

Question

Emma Aldridge · Accepted Answer

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:
https://tutor.hix.ai

Luke Alvarez · Answer

undefined Some examples of non-differentiable functions include:
1. Absolute value function (|x|)
2. Step function (Heaviside step function)
3. Modulus function (mod x)
4. Sigmoid function (logistic function)
5. The Weierstrass function
6. The Dirichlet function