A Real Function with Nonnegative Second Derivative is Convex on an Interval

Let $\mathbb{R}$ be the real numbers and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line. Let $J\subseteq\mathbb{R}$ be order-convex and such that every $t\in J$ satisfies $u<t<v$ for some $u,v\in J$, so that every point of $J$ is an interior point of $J$.

Let $g:J\to\mathbb{R}$ and $g_1:J\to\mathbb{R}$ satisfy the following, derivatives being those of Derivative at an Interior Point and well defined by Uniqueness of the Derivative at an Interior Point: for every $t\in J$ the function $g$ is differentiable at $t$ with $g'(t)=g_1(t)$, and the function $g_1$ is differentiable at $t$ with

Then for all $s,t\in J$ and every $\theta\in\mathbb{R}$ with $0\le\theta$ and $\theta\le1$, the point $(1-\theta)\,s+\theta\,t$ belongs to $J$ and