Suppose $F\subset [0,1]$ is closed. Is it true that the typical (in the sense of Baire category) function in $C^1[0,1]$ is one-to-one on $F$? If ${\underline{dim}}_{B}F<1/2$ we show that the answer to this question is yes, though we construct an $F$ with ${dim}_{B}F=1/2$ for which the answer is no. If $C_{\alpha }$ is the middle-$\alpha$ Cantor set we prove that the answer is yes if and only if $dim\left(C_{\alpha }\right)\le 1/2.$ There are $F$’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.