Using Darboux theorem for $f'$ check whether the given function $f(x)=x-\left[x\right], x\in[0,2]$ is a derivative of a function.
Please help me to solve the problem. I see that $f(x)=0=f(2)$. What to do?
Using Darboux theorem for $f'$ check whether the given function $f(x)=x-\left[x\right], x\in[0,2]$ is a derivative of a function
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calculus
real-analysis
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0What does $[x]$ mean? The integer part of $x$? – 2017-02-01
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0@FrankLu Yes, integer part – 2017-02-01
1 Answers
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Hint: Note that $$f(x)=\begin{cases} x, &0\leq x<1\\ x-1, &1\leq x<2\\ 0, &x=2\end{cases}.$$ Thus $f$ has a jump discontinuity at $x=1$.
On the other hand Darboux theorem tells you that the derivative of a function satisfies the intermediate value property. Now can you see that $f$ fails to satisfy this property?