As an example of a convex continuous function that is not everywhere differentiable, consider f ( x ) = | x | . This gives f ∗ ∗ ( x ∗ ∗ ) = sup x ( x x ∗ ∗ − − | x | ) = max ( sup x ≥ ≥ 0 x ( x ∗ ∗ − − 1 ) , sup x ≤ ≤ 0 x ( x ∗ ∗ + 1 ) ) , and thus f ∗ ∗ ( x ∗ ∗ ) = 0 on its domain I ∗ ∗ = [ − − 1 , 1 ] .