Note that every point in the interval (0,1) of the real axis satisfies your inequation. So you need only to show that the domain of f intersects with this interval.

If f is a polynomial, since every polynomial( except the constants) has a zero point, a neighborhood of 0 is covered by the domain of f, we're done.

Otherwise the power series of f has infinitly many terms, thus is an essential singular point of f. According to big picard, we're done.