Coordinate derivatives give directional Taylor expansion for convex functions ($f(x+th)=f(x)+t\nabla f(x{)}^{\mathsf{T}}h+o(t)$)

Let $f$ be a finite convex function on a neighbourhood of $x\in {\mathbb{R}}^d$, and suppose its first partial derivatives exist at $x$. For any fixed vector $h$, convexity givesThe equality uses the one-dimensional first-order Taylor expansion along each coordinate line; no continuity of second partial derivatives is assumed. Applying this upper bound to $-h$ and using $f(x)\le [f(x+th)+f(x-th)]/2$ gives the matching lower bound. Hence the displayed directional expansion holds. This supplies the expansion used in the supporting inequality for a differentiable convex function even when continuous second partial derivatives have not been postulated.