Simultaneous extension of continuous and uniformly continuous functions

The first known continuous extension result was obtained by Lebesgue in 1907. In 1915, Tietze published his famous theorem generalising Lebesgue's from the plane to general metric spaces. He constructed an explicit formula involving distance function on space. Thereafter, several authors contributed other formulas. present paper, we show that all these constructions also preserve uniform continuity, which answers a question posed St. Watson. fact, such are simultaneous for special bounded functions. Based this, refine of Dugundji constructing various (nonlinear) operators continuity as well.