Let f be a convex function defined on a convex set S in a normed linear space

Question:

Let f be a convex function defined on a convex set S in a normed linear space X. For every x0 ∈ int S there exists a linear functional g ∈ X* that bounds f in the sense that
f (x) ≥ f (x0) + g(x - x0) for every x ∈ S
Such a linear functional g is called a sub gradient of f at x0. Consider a supporting hyperplane to epi f at .x0, f .x0)).]