For instance, the absolute value function is defined and continuous on the whole real line, but its derivative behaves like a step function with a jump-discontinuity.
For some nice functions, though, such as $e^x$ or $\sin(x)$, the derivatives of course are no "worse" than the original function.
Can I say something that is typical of the derivative? Is it typically not as nice as the original function?
Best Answer
Yes, that is completely right. And inversely, integration makes functions nicer.
One way of measuring how "nice" a function is, it by how many derivatives it has. We say a function $f\in C^k$ if it is $k$ times continuously differentiable. The more times differentiable it is, the nicer a function is. It is "smoother". So if a function is $k$ times differentiable then its derivative is $k-1$ times differentiable. A function is "as nice" as its derivative if and only if its smooth (infinitely differentiable). These are functions like $\sin(x), e^x$, polynomials, etc.
Inversely, integration makes things nicer. For example integrating even a non continuous function results in a continuous function: Is an integral always continuous?