Low-temperature specific heat per site (C) of one-dimensional Hubbard model is investigated by the method of non-linear integral equations. For the half-filled case we show lim H →0 lim T →0C / T = πI0( π/ 2 U )/( 6 I1(π/ 2 U ) ), where T is temperature, H is magnetic field, U is the coupling constant, and I0 and I1 are modified Bessel functions. Although this equation yields lim T, H →0C / T = π/ 6 in the limit U →0+, the true value of lim T, H →0C / T at U = 0 is π/ 3. This means that lim T, H →0C / T is a discontinuous function of U at U = 0. This discontinuity disappears when the band is not half filled.