Summary: The taut string method is classically used in statistical applications to obtain a sparse estimation for a density given by point measurements. Mostly, a discrete formulation is employed that interpretes the data and the output as piecewise constant splines. This paper deals with the continuous formulation of this algorithm. We show that it is able to deal with continuous data as well as with discrete data interpreted as Dirac measures. In fact, any one-dimensional finite signed Radon measure is suited as input for the method. Moreover, we study the usage of tubes of nonconstant diameter. Examples indicate that such tubes can be useful in various applications. An existence and uniqueness theorem is given for the continuous formulation of the taut string algorithm with arbitrary tubes of nonnegative diameter.