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)={1−x x<02−x x≥0
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)={x2sin(1/x) if x≠00 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.
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