Let $l$ be a continuous bounded function ($l$ is not differentiable). I want to prove for $x$ large enough that $$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\frac{(x-y)^2}{2\varepsilon}}l(y)dy\leq C\frac{1}{x}. $$ Where $C$ is a positive constant, and uniformly in $\varepsilon$.
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fxed some typos and grammar issues. added the "uniformly in epsilon", as suggested by the OP's comments
leo monsaingeon
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$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\frac{(x-y)^2}{2\varepsilon}}l(y)dy\leq C\frac{1}{x}$
yassine yassine
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