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 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

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 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

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.

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 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