Bijection between multivariate functions and a nested sequence of univariate functions

Question

Is there a bijection between a sequence of univariate functions $\{\{f_{n,m}(x)\}_{m\in \mathbb{N}}\}_{n\in \mathbb{N}}$ and a sequence of multivariate functions $\{f(x,y)\}_{n\in \mathbb{N}}$. My question mostly applies to functions $f_{n,m}(x)\,:\,\mathbb{R} \to \mathbb{R}$ and $f_{n,m}(x)\,:\,\mathbb{R}^2 \to \mathbb{R}$ but I would also be curious if there are also other domains that would mess the property up if it was true.