This paper examines smooth, progressive income tax functions as an alternative to the current piecewise UK tax function. The piecewise function applies constant rates within income brackets leading to a discontinuous first derivative or marginal tax rate which can create incentives for individuals to limit their earnings. Building on Odland’s work on the Norwegian income tax function, we derive and analyse a general class of smooth tax functions that maintain progressivity while ensuring continuity in marginal rates. We find that the function Odland proposed as a replacement for the Norwegian piecewise tax function also provides a good fit for the UK piecewise tax function. Despite theoretical advantages of a smooth tax function, the implementation of such a system in the UK and elsewhere is likely to be constrained for several reasons including administrative complexity and political considerations.