Intuition behind non-continuous derivatives

Question

Although the intuition behind continuity and derivatives seem pretty obvious to me, I cannot figure out what it means to have a differentiable function with a non-continuous derivative. I'm trying to build that intuition cause I think it might help me in the problem of proving that a derivative can only have discontinuities of the second kind.

At first, I thought it might mean that the rate of growth can change as in spikes or like jump discontinuities (jumping "directly" from one value to another), however, I feel like that's pretty wrong due to Darboux's theorem.

I know I have not made this rigorous, but I am going after building my intuition now, not my rigor. Thanks in advance.

Answer 1

It might help if you have an example. http://calculus.subwiki.org/wiki/Derivative_of_differentiable_function_need_not_be_continuous

for instance. What happens here is that the function oscillates faster and faster, so the derivative jumps up and down more and more and never settles on a value as $x\rightarrow 0$.

Answer 2

Another example is just the absolute value function $f(x) = |x|$. In this case $f'(x) = -1$ for $x < 0$ and $f'(x) = 1$ for $x > 0$. The first derivative does not exist at $x=0$ and so its first derivative is not continuous at this point, but the first derivative exists everywhere else. Think of the graph of the first derivative and how it makes an obvious jump when you move from negative values of $x$ to positive.