calculus - What is the difference between differentiability of a function and continuity of its derivative?

I am sort of confused regarding differentiable functions, continuous derivatives, and continuous functions. And I just want to make sure I'm thinking about this correctly.

(1) If you have a function that's continuous everywhere, then this doesn't necessarily mean its derivative exists everywhere, correct? e.g.,

$$f(x) = |x|$$ has an undefined derivative at $x=0$

(2) So this above function, even though its continuous, does not have a continuous derivative?

(3) Now say you have a derivative that's continuous everywhere, then this doesn't necessarily mean the underlying function is continuous everywhere, correct? For example, consider

$$f(x) = \begin{cases} 1 - x \ \ \ \ \ x<0 \\ 2 - x \ \ \ \ \ x \geq 0 \end{cases}$$
So its derivative is -1 everywhere, hence continuous, but the function itself is not continuous?

So what does a function with a continuous derivative say about the underlying function?

Answer

A function may or may not be continuous.

If it is continuous, it may or may not be differentiable. $f(x) = |x|$ is a standard example of a function which is continuous, but not (everywhere) differentiable. However, any differentiable function is necessarily continuous.

If a function is differentiable, its derivative may or may not be continuous. This is a bit more subtle, and the standard example of a differentiable function with discontinuous derivative is a bit more complicated:

$$f(x) = \cases{x^2\sin(1/x) & if $x\neq 0$\\ 0 & if $x = 0$}$$
It is differentiable everywhere, $f'(0) = 0$, but $f'(x)$ oscillates wildly between (a little less than) $-1$ and (a little more than) $1$ as $x$ comes closer and closer to $0$, so it isn't continuous.