The question of the existence of a subset of a prescribed type in \(f[S]\) is studied, where \(S\) is a subset of the real line of continuum cardinality and \(f\) is assumed to be continuous or differentiable. It is shown that for continuous \(f\) the existence of the interval \([0,1]\) in \(f[S]\) is equivalent to the existence of a perfect set in this image. Moreover, the existence of such a function implies that there exist a \(C^{\infty}\) function with the same property. Some corollaries concerning images of nowhere dense sets and validity of related statements under the continuum hypothesis are shown.