Can you give me a hint how to check uniform convergence here: $$\varphi\in\mathcal{C}^\infty_0(\mathbb{R}):\quad\sup_{t\in\mathbb{R}}\left|\frac{1}{h}\left\{\varphi(t+h)-\varphi(t) \right\}-\varphi'(t)\right|\stackrel{h\to0}{\to}0$$ Pointwise is clear to me. Intuitively, this also makes sense.
2026-04-03 16:21:17.1775233277
Mollifiers: Uniform Convergence
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Hint: $$ \frac{\varphi(t+h)-\varphi(t)}{h} - \varphi'(t) = \frac{1}{h}\int_{0}^{h}\{ \varphi'(t+s)-\varphi'(t)\}\,ds $$