Question

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

Answer 1

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.