AbstractIn this note we examine certain sharpenings of the Tietze-Urysohn extension theorem. We prove, for example, that ifAis a closed subspace of a normal spaceXthen there is acontinuousextenderη:C*(A)→C*(X), whereC*(A)andC*(X)are the Banach spaces of continuous, bounded, real-valued functions onAandX, respectively. However, well-known examples show that we cannot arrange thatηis both continuous and linear, or thatηis an isometry.