As your question is not completely precise, this might not really an answer but at least it might be an occasion to modify your question in a more precise way.

If you enlarge the point of view from continuous functions to Borel functions, then the problem is easy: all Polish spaces with no isolated points are Borel-isomorphic, so if for example $x=x(t)$ with inverse $t=t(x)$ is a bijection that
preserves Borel sets you can use $F(t)=f(x(t))$ as transformation $T$ (not only linear, but also multiplication and order preserving; it does not preserve the spaces of continuous functions)

Looking only at continuous functions, you can use also a "dual" of the "space filling curve" approach: the Arnold - Kolmogorov theorem about Hilbert 13th problem. There are versions with only one external function (and, as usual, the internal functions are independent from $f$ so that the representation is really due to a continuous embedding of the $n$-dimensional cube in a larger euclidean space in such a way that continuous functions on the $n$-dimensional cube are identified with continuous functions of one real variable, the sum of the coordinates of the embedding). There are also explicitly constructed such embeddings, for example http://wissrech.ins.uni-bonn.de/research/pub/braun/remonkoe.pdf and you can easily search for more.

Edit: I have just seen in https://mathoverflow.net/questions/140859/kolmogorov-superposition-for-smooth-functions that you already know Vitushkin's theorems that kill the possibility of a general representation with differentiable functions. Even recent activity not cited in that thread confirm the "no-go" status beyond continuity, see for example http://emis.library.cornell.edu/journals/HOA/FPTA/Volume2010/287647.pdf