The main result of the paper says that for each \(k\in N\) there exists a space-filling function \(f:[0,1]\to[0,1]^k\) such that \(f\) is at most \((k+1)-\mathrm{to} -1\) at each \(y\in[0,1]^k\) (which means that card \((f^{-1}(\{y\}))\leqslant k+1\) for each \(y\in[0,1]^k\)), \(f\) is exactly \((k+1)-\mathrm{to} -1\) at a countable dense subset of \([0,1]^k\) and \(f\) is \(1-\mathrm{to}-1\) on a residual subset of \([0,1]\). The result is in a sense the best possible because from the classical theorem of Hurewicz (number 6 among the references) it follows that \(f\) is at least \((k+1)-\mathrm{to} -1\) at a countable dense set. The proof uses the generalization of so called Lebesgue partion of \([0,1]^k\) (number 10 among the references) as well as the above mentioned theorem of Hurewicz.