False. A concave down function will either arch down towards the x axis or have arched from the xaxis to some asymptotic ceiling. In either case it must cross the x-axis somewhere, assuming it must be defined for all real X from -inf to inf.
@Above: none of those functions is strictly positive or strictly negative for all x.
Edit: Actually, would a periodic, piecewise function work? Like having sin(x) for pi/6<=x<pi/3 repeated throughout. It would be discontinuous, but yea.