Assume that $k \in \mathbb{N}$ and $f : \mathbb{R}^d \rightarrow [0,\infty)$ is lower semicontinuous, i.e. $f(x) \leq \liminf_{y \rightarrow x} f(y)$ for all $x \in \mathbb{R}^d$.

Does there exist an increasing sequence of $k$-times continuously differentiable functions $(g_n)_n \subset C^k(\mathbb{R}^d ; [0,\infty))$ that converges pointwise to $f$, i.e. $g_n(x) \leq g_{n+1}(x)$ for all $x \in \mathbb{R}^d$ and $n \in \mathbb{N}$ as well as $\lim_{n \rightarrow \infty} g_n(x) = f(x)$ for all $x \in \mathbb{R}^d$?

My intuition would be yes, since we could tile $\mathbb{R}^d$ into dyadic cubes, take for the center of every cube the minimum of the values of $f$ on neighboring cubes and then interpolate between these center points with nice $C^\infty$ functions. Unfortunately this approach sounds very technical to me and I'm wondering whether there is something more elegant.