Let \(C[a,b]\) be the space of continuous real-valued functions on \([a,b]\) with uniform norm \(\|\;\|\). For \(f\) in \(C[a,b]\), let \(B_ n(f)\) denote the best uniform approximate from the set of algebraic polynomials of degree \(n\) or less to \(f\). \(E_ n(f)\) is the number of the extremal points of \(f-B_ n(f)\). We define the local Lipschitz constant for \(f\) by \(\lambda^ \ell_ n(f)=\lim_{\delta\to+0}\sup\{\| B_ n(f+\varphi)-B_ n(f)\|/\|\varphi\|;\;0<\|\varphi\|<\delta\}\). Then Angelos, Henry, Kaufman, Kroó and Lenker showed that if \(E_ n(f)=n+2\) for all sufficiently large \(n\), \(\lim^ \lambda_ n(f)=\infty\). The authors study the behavior of \(\lambda^ \ell_ n(f)\) by using Kolushov polynomials. First they characterize \(\lambda^ \ell_ n(f)\) for fixed \(n\) in terms of Kolushov polynomials and second show that \(\lim \lambda^ \ell_ n(f)=\infty\) holds in the more general case.