In this paper, we investigate the universal approximation property of deep, narrow multilayer perceptrons (MLPs) for $C^1$ functions under the Sobolev norm, specifically the $W^{1, \infty}$ norm.

We present a geometric framework that reduces the problem of finding the minimum width for universal approximation by deep, narrow MLPs to a dimension-based function w(d_x,d_y) yielding tight upper and lower bounds under the uniform norm.