Does derivative have to be piecewise continous

[causalset]

I know the derivative does not have to be continous, due to couter-example $f(x)=x^2 \sin (1/x)$. But does derivative still have to be PIECEWISE continuous? If not, is there some weaker statement that is still true?

 

[g_edgar]

the derivative has the Darboux property... http://planetmath.org/encyclopedia/DarbouxsTheorem.html"

 

[causalset]

the derivative has the Darboux property... http://planetmath.org/encyclopedia/DarbouxsTheorem.html"

Are you saying that derivative does NOT have to be piecewise continuous, or are you saying you simply don't know one way or the other?