The variable exponent De Giorgi class and the variable exponent Hölder continuity

For a bounded domain $ \Omega $ in $ \mathbb{R}^{n} $ ($ n \!\ge\! 2 $), we prove that the variable exponent De Giorgi class $ \mathscr{B}_{m(x)} (\Omega, M, \gamma, \gamma_1, \delta) $ is a subset of the variable exponent Hölder space $ C^{0, \alpha(x)}_{loc}(\Omega) $. It gives an improvement for the earlier result, which states that $ \mathscr{B}_{m(x)} (\Omega, M, \gamma, \gamma_1, \delta)\subset C^{0, \alpha}_{loc}(\Omega) $, where $ \alpha $ is a constant. This result can be applied to get $ C^{0, \alpha(x)}_{loc}(\Omega) $ regularity for certain solutions of variational problems and nonlinear elliptic equations with variable exponents.