Construct a set of functions $\{g_\epsilon(x) \}$, such that for every $\epsilon > 0, \; g_\epsilon(x)$ is infinitely differentiable and $$ g_\epsilon \rightarrow f,$$ where $f(x) = |x|,$ in the sup norm as $\epsilon \downarrow 0$.
Sequence of smooth functions converging to the modulus function
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calculus