# Non Differentiable Functions

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

## Graphical Meaning of non differentiability.

Which Functions are non Differentiable?
Let f be a function whose graph is G. From the definition, the value of the derivative of a function f at a certain value of x is equal to the slope of the tangent to the graph G. We can say that f is not differentiable for any value of x where a tangent cannot 'exist' or the tangent exists but is vertical (vertical line has undefined slope, hence undefined derivative).
Below are graphs of functions that are not differentiable at x = 0 for various reasons.
Function f 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 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

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.

## Examples with Solutions

Analytical Proofs of non differentiability

Example 1: Show analytically that function f defined below is non differentiable at x = 0.

f(x) = \begin{cases} x^2 & x \textgreater 0 \\ - x & x \textless 0 \\ 0 & x = 0 \end{cases}

Solution to Example 1
One way to answer the above question, is to calculate the derivative at x = 0. We start by finding the limit of the difference quotient. Since function f is defined using different formulas, we need to find the derivative at x = 0 using the left and the right limits.

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 .