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Let $\{g_n\}$ be a sequence of concave functions defined on $\mathbb{R}$ and set

$$\lambda_n(x)=\lim_{\Delta x\to 0+}\frac{g_n(x+\Delta x)-g_n(x)}{\Delta x}$$

Assume there exists a concave function $g$ s.t. for every $x\in\mathbb{R}$,

$$g_n(x)\to g(x),~ n\to\infty$$

Could we show that $\lambda_n(x)$ is convergent for every $x\in\mathbb{R}$? Thanks for the reply!

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  • $\begingroup$ The question to ask is, is $\lim_{\Delta x\to0}\lim_{n\to\infty}f_n(\Delta x)=\lim_{n\to\infty}\lim_{\Delta x\to0}f_n(\Delta x).$ If so, then $\lim_{n\to\infty}\lambda_n(x)=\lambda(x)=g^\prime(x)$. $\endgroup$
    – user62675
    Commented May 11, 2014 at 5:45
  • $\begingroup$ I mean if we have a convergent sequence of concave functions, could be possible that its right-derivative converges? $\endgroup$
    – CodeGolf
    Commented May 11, 2014 at 6:22
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    $\begingroup$ No: Let $h(x) = -x$ for $x \geq 0 $ and $0$ for $x < 0$. Then $g_n(x) = h(x-1/n)$ converge to $h$ but $\lambda_n(0) = 0$ for all $n$ while $\lambda(0) = -1$ for $h$. $\endgroup$
    – Connor Mooney
    Commented May 16, 2014 at 13:11

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