设 $f(x)\in R[0,1], F(x)=\int_0^x f(t)\mathrm{ d} t, DF(x)=\varlimsup_{h\to 0}\frac{F(x+h)-F(x)}{h}$. 求证: $$\begin{aligned} DF(x)\in R[0,1]\mbox{且} \int_0^1 DF(x)\mathrm{ d} x=\int_0^1 f(x)\mathrm{ d} x. \end{aligned}$$

zhangzujin

设 $\displaystyle f(x)\in R[0,1], F(x)=\int_0^x f(t)\mathrm{ d} t, DF(x)=\varlimsup_{h\to 0}\frac{F(x+h)-F(x)}{h}$. 求证: $$\begin{aligned} DF(x)\in R[0,1]\mbox{且} \int_0^1 DF(x)\mathrm{ d} x=\int_0^1 f(x)\mathrm{ d} x. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

flyy

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