This paper investigates the fractal dimension of continuous and differentiable curves through the construction of function sequences defined on the unit closed interval \([0,1]\) . Sequences composed of piecewise linear functions and their Box dimensions are analyzed. Under appropriate conditions, it is shown that the Box dimension of the graphs in \(\mathbb {R}^2\) exceeds the Box dimension of the corresponding intersection set in \(\mathbb {R}\) by one, when infinitely many lines intersect the entire interval \([0,1]\) . Additionally, in cases where the lines intersect only a portion of \([0,1]\) , an upper bound for the Box dimension has been established using analogous methods.
