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