# Non Differentiable Functions

Questions on the differentiability of functions with emphasis on piecewise functions are presented along with their answers.

Function g below is not differentiable at x = 0 because there is no tangent to the graph at x = 0.(try to draw a tangent at x=0!)
Function h below is not differentiable at x = 0 because there is a jump in the value of the function and also the function is not defined therefore not continuous at x = 0.
Function j below is not differentiable at x = 0 because it increases indefinitely (no limit) on each sides of x = 0 and also from its formula is undefined at x = 0 and therefore non continuous at x=0 .
Function k below is not differentiable because the tangent at x = 0 is vertical and therefore its slope which the value of the derivative at x =0 is undefined.
Theorem: If a function f is differentiable at x = a, then it is continuous at x = a
Contrapositive of the above theorem: If function f is not continuous at x = a, then it is not differentiable at x = a.
Common mistakes to avoid: If f is continuous at x = a, then f is differentiable at x = a. NOTE: Although functions f, g and k (whose graphs are shown above) are continuous everywhere, they are not differentiable at x = 0. Analytical Proofs of non differentiability
f(x) = \begin{cases}
x^2 & x \textgreater 0 \\
- x & x \textless 0 \\
0 & x = 0
\end{cases}
f'(x) = \lim_{h\to\ 0} \dfrac{f(x+h) - f(x)}{h}
On the left of x = 0 (x < 0), the derivative is calculated as follows
f'(0) = \lim_{h\to\ 0^-} \dfrac{f(0+h) - f(0)}{h} = \lim_{h\to\ 0} \dfrac{ -h - 0}{h} = -1
On the right of x = 0 (x > 0), the derivative is calculated as follows
f'(0) = \lim_{h\to\ 0^+} \dfrac{f(0+h) - f(0)}{h} = \lim_{h\to\ 0} \dfrac{h^2 - 0}{h} = \lim_{h\to\ 0} h = 0
The limits to the left and to the right of x = 0 are not equal therefore f '(0) is undefined and function f in not differentiable at x = 0. More on Continuous Functions in Calculus Continuity Theorems and Their use in Calculus Questions on Continuity with Solutions. Home Page |