This paper is concerned with temporal planning relying on CSP-based functional representations. These powerful representations are today mostly restricted to the use of piecewise constant functions ranging over finite domains. We are proposing here an extension that brings a significant enhancement in the expressiveness of the representation, towards handling continuous change. This extension consists mainly in allowing piecewise linear functions over continuous domains. We have studied this extension and are currently implementing it in the IXTET planner. However this extended representation is not specific to IXTET; it can be useful to most temporal planners. We show how IXTET syntax, planning algorithm and control can be simply extended to this class of functions. We then consider the more significant modifications required in the two constraints managers for handling temporal and atemporal CSPs.