Can there be a set whose Housdorff dimention gradually changes?

For instance, a set of real numbers contained in an interval, whose Housdorff dimention is 0 at the beginning and 1 closer to the end, and changes without jumps?

Can there be a set whose Hausdorff dimension gradually changes?

For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, and changes without jumps?