Is a function differentiable at all points that it is continuous?

1 Answer
Mar 1, 2016

Answer:

No. Here are 3 examples.

Explanation:

Example 1
#f(x) = absx# is continuous but not differentiable at #x=0#.

(The left and right derivatives are not equal -- there is no tangent line.)

graph{y=absx [-2.75, 2.724, -0.876, 1.862]}

Example 2
#f(x) = root3x# is continuous but not differentiable at #x=0#.

(#f'(x) = 1/(3root3(x^2))# does not exist at #x=0#. In fact,)
(#lim_(xrarr0) abs(f'(x)) = oo# -- the tangent line is vertical.)

graph{root(3)x [-1.596, 1.441, -0.964, 0.555]}

Example 3
#f(x) = root3(x^2)# is continuous but not differentiable at #x=0#.

(#f'(x) = 2/(3root3x)# does not exist at #x=0#. In fact,)
(#lim_(xrarr0) abs(f'(x)) = oo# -- the tangent line is vertical.)

graph{x^(2/3) [-1.82, 1.597, -0.343, 1.366]}

I like this third example because it is also an example of a function whose minimum occurs at a critical point at which the derivative does not exist.