Let $f : X^3 \rightarrow X$.

If $X$ is $\mathbb Z$, then there will be a couple of functions $g,h$ from $\mathbb Z^2$ to $\mathbb Z$ that satisfies $f(x,y,z) = g(h(x,y),z)$ since there is a bijection $h :\mathbb Z^2 \rightarrow \mathbb Z$.

However, If $X$ is $\mathbb R$, then there are no continuous bijection from $\mathbb R^2$ to $\mathbb R$.

My question is : Is there any continuous function $f : \mathbb R^3 \rightarrow \mathbb R$ that can't be represented by composition of continuous functions $g_i = \mathbb R^2 \rightarrow \mathbb R$? And similar questions for $f : \mathbb R^n \rightarrow \mathbb R$.

Sorry, I'm not sure about my tags.