The FEAST eigensolver uses approximate spectral projection to compute all of the eigenvalues of a matrix that are located in a bounded region of the complex plane. We develop a continuous analogue of the FEAST algorithm that manipulates differential operators rather than matrices. By exploiting the continuum structure of the operator rather than the structure of intermediate discretizations, we obtain an efficient, spectrally accurate eigensolver that does not suffer from difficulties, such as non-normality and ill-conditioning, associated with large spectral discretizations of differential eigenvalue problems. We demonstrate that this eigensolver is particularly well-suited for computing high-frequency eigenmodes of Sturm-Liouville operators.