Consider a real valued function g on an open interval $(a,b)$ which is the derivative of a function continuous on $[a,b]$ at each point of $(a,b)$. The function $g$ has the intermediate value property, so a monotone $g$ will have to be continuous, a general $g$ cannot have simple discontinuities, etc. With such constraints how badly can a derivative behave in terms of continuity, can it get much worse than the derivative of say, $x^2 sin(1/x) \sin(1/x) $?
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Consider a real valued function g on an open interval (a,b) $(a,b)$ which is the derivative of a function continuous on [a,b] $[a,b]$ at each point of (a,b). g $(a,b)$. The function $g$ has the intermediate value property, so a monotone g $g$ will have to be continuous, a general g $g$ cannot have simple discontinuities, etc. With such constraints how badly can a derivative behave in terms of continuity, can it get much worse than the derivative of say, $x^2 sin(1/x)sin(1/x) $? |
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Examples of badly behaved derivativesConsider a real valued function g on an open interval (a,b) which is the derivative of a function continuous on [a,b] at each point of (a,b). g has the intermediate value property, so a monotone g will have to be continuous, a general g cannot have simple discontinuities, etc. With such constraints how badly can a derivative behave in terms of continuity, can it get much worse than the derivative of say, x^2 sin(1/x)?
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