Questions with answers on the differentiability of functions with emphasis on piecewise functions.
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 differntiable 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: 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 diffenrentiable at x = 0.
Analytical Proofs of nondifferentiability
Example 1: Show analytically that function f defined below is non differentiable at x = 0.
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.
Derivative: f' (x) = limh→0[f(x+h) - f(x)]/h
at x= 0; f '(0)(left) = limh→0[f(0+h) - f(0)]/h = limh→0[- h - 0]/h = -1
at x= 0; f'(0)(right) = limh→0[f(0+h) - f(0)]/h = limh→0[h 2 - 0]/h = limh→0[h 2 - 0]/h = limh→0h = 0
The limits to the left and to the right 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.