Let 聿= be a metric space and let 汍 be a positive real number. Then a function f: X↙Y is defined to be an 汍-map if and only if for all y﹋Y, the diameter of f^-1(y) is at most 汍. In Theorem 10 we will give a new proof for the following well known fact: if 聿 is totally bounded, then for all 汍 there exists a finite number n and a continuous 汍-map f_汍: X↙Rⁿ (here Rⁿ is the usual n-dimensional Euclidean space endowed with the Euclidean metric). If 汍 is ※small§, then f_汍 is ※almost injective§; and still exists even if 聿 has infinite covering dimension (in this case, n depends on 汍, of course). Contrary to the known proofs, our proof technique is effective in the sense, that it allows establishing estimations for n in terms of 汍 and structural properties of 聿.