Let $\mathcal{C}^0([a,b],\mathbb{R})$ be the space of all continuous functions $f:[a,b]\rightarrow\mathbb{R}$ and $\mathcal{C}^\infty([a,b],\mathbb{R})$ the subspace of all smooth functions. Define $f\leq g:\Leftrightarrow(\forall x: f(x)\leq g(x))$ and $f\ll g:\Leftrightarrow(\forall x: f(x)< g(x))$. Is the following true:

$$\forall f,h\in\mathcal{C}^0([a,b],\mathbb{R}) \;\exists g\in\mathcal{C}^\infty([a,b],\mathbb{R}): f\ll h\Rightarrow f\leq g\leq h,$$ i.e. between any two continuous functions $f\leq h:[a,b]\rightarrow\mathbb{R}$, there exists a smooth function $g$?

Does this yield that $(\mathcal{C}^r([a,b],\mathbb{R}),\leq)$ is not a lattice, for any $r\in\mathbb{N}$.

I haven't found this result anywhere in my calculus books, so I'd appreciate any references.

Let $\mathcal{C}^0([a,b],\mathbb{R})$ be the space of all continuous functions $f:[a,b]\rightarrow\mathbb{R}$ and $\mathcal{C}^\infty([a,b],\mathbb{R})$ the subspace of all smooth functions. Define $f\leq g:\Leftrightarrow(\forall x: f(x)\leq g(x))$ and $f\ll g:\Leftrightarrow(\forall x: f(x)< g(x))$. Is the following true:

$$\forall f,h\in\mathcal{C}^0([a,b],\mathbb{R}) \;\exists g\in\mathcal{C}^\infty([a,b],\mathbb{R}): f\ll h\Rightarrow f\leq g\leq h,$$ i.e. between any two continuous functions $f\ll h:[a,b]\rightarrow\mathbb{R}$, there exists a smooth function $g$?

Does this yield that $(\mathcal{C}^r([a,b],\mathbb{R}),\leq)$ is not a lattice, for any $r\in\mathbb{N}$ (I'd like to construct a function $g$ between $|x|$ and supposed $x\vee-x$).

I haven't found this result anywhere in my calculus books, so I'd appreciate any references.