Does derivative have to be piecewise continous

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SUMMARY

The derivative of a function does not need to be piecewise continuous, as demonstrated by the counter-example \( f(x) = x^2 \sin(1/x) \). This function's derivative possesses the Darboux property, which indicates that it can take on all intermediate values within any interval, despite not being continuous. The discussion clarifies that while derivatives may not be piecewise continuous, they still exhibit certain properties that can be explored further.

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