  We identify a class of continuous compactly supported functions for which the
known part of the Gabor frame set can be extended. At least for functions with
support on an interval of length two, the curve determining the set touches the
known obstructions. Easy verifiable sufficient conditions for a function to
belong to the class are derived, and it is shown that the B-splines $B_N, N\ge
2,$ as well as certain "continuous and truncated" versions of several classical
functions (e.g., the Gaussian and the two-sided exponential function) belong to
the class. The sufficient conditions for the frame property guarantees the
existence of a dual window with a prescribed size of the support.
