Assume one has the function $g(x)=f(x)1_{[a,b]}(x)$ with $f\in C(\mathbb{R})$. Can one finds a sequence of test functions $\psi_n\in C_c^\infty(\mathbb{R})$ such that
$$\psi_n\rightarrow g, \;\;\;\;\; with \; \; \; \; \; \psi_n'=0.$$
2026-04-13 02:34:45.1776047685
Approximation with null derivative test functions
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The function $g$ can be approximated with piecewise constant functions $\gamma_n$.
Take a mollifier $\rho_n$ and set $\psi_n=\gamma_n*\rho_n.$ Then $\psi_n \to g*\delta=g.$