In this paper, the author proves that if the dual X* of X is weakly locally uniformly convex and the convex function f is continuous on X, then there exist two sequences \(\{f_{n}\}_{n=1}^{\infty}\) and \(\{g_{n}\}_{n=1}^{\infty}\) of continuous functions on X** such that (1) fn(x) ≤ fn+1(x) ≤ f(x) ≤ gn+1(x) ≤ gn(x) whenever x ∈ X; (2) the two convex functions fn and gn are Gâteaux differentiable on X; (3) fn → f and gn → f uniformly on X. Moreover, if the function f is coercive on X, then (1) fn and gn are two w*-lower semicontinuous convex functions on X*; (2) \(\text{epi}\;f_{n}=\overline{\text{epi}\;f_{n}\cap(X\times R)}^{w^{\ast}}\) and \(\text{epi}\;g_{n}=\overline{\text{epi}\;g_{n}\cap(X\times R)}^{w^{\ast}}\) .
