If $[a,b)$ ($a
Right derivative of a continuous nondecreasing map
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calculus
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1Consider $f(x)=\sqrt x$ on $[0,1)$ at $x=0$. – 2017-02-19
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0I will reformulate. Take the open interval $(a,b)$. – 2017-02-19
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0Do you know the cantor function? – 2017-02-19
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0I know it is continuous and increasing and has a derivative almost everywhere. Am I right? But does it have a right derivative everywhere? – 2017-02-19
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1sorry. Please forget the cantor function. That makes thing unnecessarily complicated. Consider $f(x) = \sqrt[3]{x}$ on any open interval containing $0$. – 2017-02-19
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0thanks. if one allows the right derivative to take on values in the extended real line, then the claim might hold true... – 2017-02-19
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0I know no useful instance in which one consider $\pm \infty$ as a valid derivative. Notice there is result that a continuous and monotonous function is almost everywhere differentiable. – 2017-02-19
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0the result says that monotonic functions (not necessarily continuous) are almost everywhere differentiable. – 2017-02-19