A function \(h:[0, 1]\to \mathbb{R}\) is said to be a polygonal function for \(f\) if there is a partition \(\{0= a_0< a_1<\cdots< a_m= 1\}\) of \([0, 1]\) such that \(h\) agrees with \(f\) at each partition point and is linear on the intervening closed intervals. Points \(a_0,a_1,\dots, a_m\) (\((a_0,h(a_0)),(a_1, h(a_1)),\dots, (a_m,h(a_m))\)) are called nodes (vertices) of \(h\). The maximum distance between consecutive nodes (vertices) is called mesh of \(h\), \(\text{mesh}(h)\) (graph-mesh of \(h\), \(\text{graph-mesh}(h)\)). A function \(f\) is said to be polygonally approximable, if there is a sequence \(\{h_n\}\) of polygonal functions for \(f\) such that \(\lim_{n\to\infty} h_n(x)= f(x)\) for every \(x\in [0,1]\) and \(\lim_{n\to\infty} \text{mesh}(h_n)= 0\). If \(\text{graph-mesh}(h_n)\) replaces \(\text{mesh}(h_n)\) then we obtain the notion of a strongly polygonally approximable function. A function \(f\) is said to be universally polygonally approximable -- UPA (strongly universally polygonally approximable -- SUPA) if for every dense subset \(D\) in \([0, 1]\) there is a sequence \(\{h_n\}\) of polygonal functions for \(f\), having nodes in \(D\cup\{0, 1\}\) which polygonally approximates \(f\) (strongy polygonally approximates \(f\)) on \([0, 1]\). The authors give characterizations of introduced classes of functions UPA and SUPA, and investigates some of their properties. E.g., it is shown that the following statements are equivalent: (i) \(f\) is Baire one, Darboux, and quasi-continuous; (ii) \(f\) is Darboux and UPA; (iii) \(f\) is SUPA. An example shows that the uniform limit of UPA functions, need not be UPA. The paper was motivated by the paper [\textit{S. J. Agronsky}, \textit{J. G. Ceder} and \textit{T. L. Pearson}, Real. Anal. Exch. 23, No. 2, 421-430 (1997; Zbl 0943.26004)].