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 \( x = 0 \).
\( f(x) = \begin{cases}
x^2 & x > 0 \\
- x & x < 0 \\
0 & x = 0
\end{cases}
\)
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 \( x = 0 \) (\( x \lt 0 \)), 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 and therefore \( f '(0) \) is undefined and hence function \( f \) in not differentiable at \( x = 0 \).
The graph of function \( f \) solved in this example is shown below and it is easy to note that no tangent can be drawn at \( x = 0 \) and hence \( f \) is not differentiable at x = 0.
