Approximate continous function with linear growth condition by Lipschitz function

Question

Suppose a continuous function $f(u)<K(1+|u|)$ for some positive number $K$. How can we find a sequence of Lipschitz functions $f_{n}$ that converge to $f$ uniformly on $\mathbb{R}$. If we require $f(u)<K(1+|u|^{\gamma})$ for $\gamma>1$, then the uniform convergency would be changed to just convergency by point limit? My idea is to use the cutoff function to approximate it. But I do not know how to prove the uniformly convergency using the assumption on $f$.

Answer 1

I would try the following. For $n>K$, consider the function $f_n(x)=\sup_{u\in\mathbb R} (f(u)-n|u-x|)$. Then check that

• $f_n$ is well defined (supremum is attained and is finite)
• $f_n$ is Lipschitz
• $f_n\ge f$
• $f_n\to f$ pointwise ... uniformity is not so clear.

In fact, I don't believe in uniform approximation. The function $f(u) = \sin u^2$ obviously satisfies the given bound, but for any Lipschitz function $g:\mathbb R\to\mathbb R$ we have $\sup|f-g|\ge 1$.