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I was wondering whether there exists a sequence $\varphi_n$ of continuous bounded functions/continuous functions vanishing at infinity on $\mathbb R$ that converge pointwise to the identity $x \mapsto x?$

I feel that continuous and bounded functions should work, as one can perhaps successively approximate the identity from the $x$-axis, i.e.

we take $\varphi_n(x) =x 1_{[-n,n]}+n1_{[-n,n]^C}.$

I am not so sure about continuous and vanishing at infinity, though.

In particular, I would like to have that $\left\lvert \varphi_n(x) \right\rvert \le C \left\lvert x \right\rvert$ uniformly in $n$ for some fixed constant $C.$

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You can even do it with compactly supported continuous functions. Take for example

$$ \varphi_n (x) = \begin{cases} x, & x\in [-n,n], \\ 2n-x,& x\in (n,2n], \\ -2n-x,& x\in [-2n, -n), \\ 0,& \vert x \vert > 2n. \end{cases} $$

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